Fixture Pan for 3/8 drywall - do they exist?
If it's a non-loadbearing wall, you can safely drill 1/4 inch deeper into the stud and then chisel out in-between.

Pancake boxes are already at a minimum size for connections, a 3/8 box couldn't be rated for connections by the NEC.  Each connection needs 2 cubic inches of volume ( 1 cu-in per wire), so the 4inch diameter by 1/2 inch pancake box is rated at 6 cu-in, just enough for one 14/2 wire.  12/2 or a splice would not be permitted in that box.