So, it would seem that Chisel uses some kind of larger texture (CTM perhaps?) for some of its blocks, and this texture is split up into 4 parts, like so:

(For reference, this is Cobblestone - Arrayed Bricks from Chisel (in Sphax BDCraft resourcepack))

As a result of this 2x2 repeating pattern for blocks, when you put Cobblestone Arrayed Bricks in the Framing table with a framed drawer, and put it into the panels slot, this is what happens:

It manages to take all 4 textures for the same block, squeezes them into one block texture, and uses that instead. As a result, I cannot use it for any walls containing Chisel bricks, because that is what it looks like, it does not fit in at all.
Hopefully there is a workaround for this or it is fixed soon. The framed drawers are a really good idea and I hope it can be used in even more implementations