CINCINNATI—Dan and Brian Busken, of the family-owned Busken Bakery, pulled into the parking lot of a Frisch's Big Boy on a recent morning for a clandestine mission: They donned ski masks with fake mustaches attached, placed a ladder next to the restaurant's namesake statue and cloaked Big Boy in a Busken apron.

To top things off, they encased the statue's trademark burger in a Busken pie box. They then posted the maneuver on YouTube and Facebook in a video complete with fake explosions and flashes of lightning.

"I...