During this period, Subway has continued to expand to more than 41,000 restaurants in 106 countries—making it the largest restaurant chain in the world.

In the nearly four decades since co-founding Subway , CEO Fred DeLuca's seen a seismic shift in the restaurant industry as consumers gravitate toward fast-casual restaurants and shift toward healthier eating.

DeLuca: Actually not much thought goes into producing the smell. It's kind of natural. Here there's a big emphasis on fresh vegetables and fresh baked bread so the most likely thing you're smelling are those two items.

There's a distinctive Subway smell. Tell me about the Subway smell — the importance of it, how you produce it and how much thought goes into it.

Jared Fogle, who became a Subway spokesperson about 16 years ago after losing 245 pounds on a Subway diet, also spoke with CNBC. He still eats at the chain at least two to three times per week and travels nearly 200 days per year with Subway around the world.

DeLuca sat down with CNBC to talk about these controversies, his solution to the minimum wage debate and the company's expansion plans.

This year, DeLuca's steered the company through a couple of controversies—notably public backlash after word spread the company was phasing out azodicarbonamide (AZO), a chemical found in yoga mats, shoe rubber and synthetic leather, from its breads. Just last week, CNN published a report that Subway led the fast food industry in underpaying employees.

You've continued to expand despite having many restaurants in the U.S. already, adding more than 2,700 last year. What are your expansion plans moving forward?

DeLuca: If you go backwards, in the past 10 years, we've more than doubled. The international component has grown enormously. ... We were kind of teeny internationally 10 years ago, and now we're opening more stores outside of the U.S. than in the U.S. So I think this pace of adding about 2,500 to 3,000 stores per year will probably continue on into the future. We might have some lower years, some higher years.

Read MoreSkip the tip, restaurant CEO says

So, If you project out maybe 10 years from now, I think it's fair to say we'd have somewhere in the range of 65,000 stores worldwide.

Where are you expanding most heavily right now?

DeLuca: U.S., of course, is No. 1. It's a very big country. Internationally in places, like Brazil, the U.K, Mexico, Russia, India, China.

Has the Russian-Ukrainian conflict impacted your plans?

DeLuca: No, I think it's kind of recent. I'm sure it's going to be disruptive to some degree.

The expansion comes amid weak growth in the QSR (quick-service restaurant) segment. Traffic's roughly flat with average check driving the growth. Why choose to expand so heavily during this low-growth environment?

It's not really a choice to expand during any given period because the actual act of opening a store is really a long-term event. In a typical situation, it's going to take pretty close to a year to get your location in, get your permits and then get open.

Right now for instance, we've got about 4,000 new franchisees in the pipeline—a huge number. We can't just turn on this pipeline and turn it off when we want.

Last year, you announced you would be phasing out AZO and completed the phaseout this year. What drove you to do this and will you be looking at any other controversial ingredients in your stores?

DeLuca: I don't think there was a specific target. It was more of getting to a common core of ingredients. We're in a lot of big countries so every country has their own rules—just like every state does. We decided 10 years ago that we were going to do a couple things. One was to work to simplify the ingredients. Another one was to work to get a common core of ingredients. So if it's approved in Europe, Australia and here, OK let's include it everywhere. AZO was one of the ones that was not in the common core.

Were you surprised at all by the controversy?

DeLuca: I was. First of all when it came out, I didn't even know what it was. There are so many ingredients that are approved for use. You can't be an expert on all of them. I had to say "What is this?" That was my first question. Then I learned that this is common in baking. Everyone uses it. It's approved, and I thought, "What is the big deal about this?" Sometimes people go in for publicity stunts to advance their own causes. I personally don't know why there was so much pickup on this.

How would you see the impact of a potential federal minimum wage increase? Are you concerned at all that the federal minimum wage could go up?

DeLuca: I'm not concerned. I know our stores owners are concerned.

When I started in the business, the minimum wage was $1.25. I've seen an enormous number of wage increases. Basically it applies evenly to everyone in the business. This increase would impact Subway plus every other competitor so it would not put any brand at a particular disadvantage. It might have a slight impact on consumers because what's going to happen is a wage increase will happen and all the restaurant owners will have to recoup that somehow, usually through a price increase. It might make eating out at restaurants have a little bit less competitive advantage compared to supermarkets.

Read MoreLandry's CEO: Diners will pay more

Over the years, I've seen so many of these wage increases. I think it's normal. It won't have a negative impact hopefully, and that's what I tell my workers. I always have whenever we come across these things. I personally think that if I were in charge of the government, I would index the minimum wage to inflation so that way everybody knows what they can count on. The employees know they're going to get increases on a regular basis. The management knows that they're going to have to pay a little bit more with inflation. It just seems much more sensible and fair to me. I don't know why it hasn't been done like that. I would do it that way because in the long, long run it's going to approximate the change in inflation.

Last week, CNN put out a report naming Subway as the fast food industry leader to underpaying workers, according to government data. What is your response to this?

DeLuca: I saw that, and I think that the headline is not quite accurate. When I read it, I said "Oh, it looks like they're writing that our wages are lower than the wages of other places." When I read through it, I saw they were talking about wage violations at the store level. So a couple of thoughts: First of all, the fact that we have so many stores has an impact on how many violations there are. If we had 5,000 stores, there would be a smaller number than if we had 25,000. ... The vast majority of our owners are doing the right thing but some are not. I would say this. We, as a company, realize that some of our owners have not done the right thing.

I can't tell you exactly why, but I think we have a lot of first-time business people that enter into business in Subway, and they might not be as sophisticated in what to do.

So a couple years ago, maybe three or four years ago, we started working closely with the Department of Labor to partner with them to help educate our owners on the right thing to do. The people who come to work deserve to be paid properly and there's no excuse. I could understand someone making a small error but sometimes people make systematic errors and that's not right.

We talked about restaurant growth before that's been really driven by the fast-casual space, which offers premium ingredients and a higher level of service. How have you seen this impact? And how have you changed your own business in response to the fast-casual growth?

DeLuca: I think the fast-casual space will probably continue to increase. Over the years, competition has increased enormously with a lot of variety, a lot of different brands and of course people eat at fast-casual spaces too.

There are some of our owners who think we should try to be like fast casual. I think we have a pretty straightforward business model here. We're trying to provide really affordable food here for the average person and good healthy food.

It's straightforward. You come up, talk to the sandwich artist. You get what you want, but we do it more, I think, in a fast food manner. But I think our system that we started way back in the last century is a system that a lot of fast casuals are starting to go with—more personalized service, the open kitchen. People get to see their food.

In a sense, we're way ahead of the times, but I wouldn't classify us as fast casual. I don't think our job is to try to be fast casual, but it's definitely a growing segment.

—By CNBC's Katie Little.