We always worried that it sent a bad message to have the city Department of Education headquartered in the Tweed Courthouse building. But even we’re shocked to see the city schools chancellor goosing the payroll in a way that Boss Tweed would envy.

As Selim Algar reports in Wednesday’s Post, pay has more than doubled for DOE Central Administration staff in the de Blasio era. Indeed, it’s set to nearly triple by next year — from just over $4 million in 2013 to $11.4 million in 2018.

Why? Chancellor Carmen Fariña is vastly increasing the size of the central bureaucracy, apparently on the theory that it’s the best way to improve your child’s classroom education.

The chancellor may well believe that — after all, she spent her career in the city system, and she believes in it. And never mind that reducing the central schools bureaucracy has been the goal of local education reformers for all those same decades.

This is hardly the first sign of Fariña’s faith in free spending as the route to educational excellence.

Just weeks ago, The Post’s Susan Edelman and Bruce Golding exposed how Fariña’s Renewal Schools program has proved a jackpot for outside consultants, paid under contracts that total around $40 million a year — on top of the $8.5 million payroll for 72 DOE bureaucrats dedicated to the program.

One consultant, Laura Kotch, had co-authored a book with Fariña. She was earning $1,200 a day — while Sandra Kase, another of the chancellor’s friends, collected $1,400 a day and over $167,000 for all of 2016.

All this when Renewal is failing badly at turning around dozens of low-performing city schools.

Usually, when we rage about how the school system is run to serve the adults, not the students, our target is the teachers union. Sadly, Fariña is proving that management can be every bit as sordid.