The city directory for 1958 lists for 2631 Bissonnet (the Marquis II location): "The Big Donut Inc." Don't know for sure if it was connected but, in the 1950's, there was a chain of drive-through donut shops that were called "Big Donut Drive-In". The Los Angeles location looked like this:
I think it's still there (in L.A.) - but called Randy's Donuts now. There were apparently about 9 other Big Donut Drive-Ins in the 1950's, but I don't know whether there was ever one in Houston. Maybe "The Big Donut Inc." was a copycat company.
I finally got back to the library to do a little more research - though I jumped around a bit because a lot of the city directories were off the shelf. I'll post when I've updated my notes.
I found the picture.