When I do build hinged doors I make them frameless to give the same type of viewing as a double slider.
I just make a 1/4" thick frame that sits on the face of the cage without cutting into the viewing area. Then I have a piece of acrylic cut that sits inside of this frame. This smaller face frame allows for the use of flush mount hinges. They do make offset hinges but they are harder to find and are odd sizes. The offset hinges allow you to skip the smaller frame.
Acylic hinges are then screwed into the wood side and solvent-welded onto the acrylic side.
If you look at the picture below you can see the main, structural face frame, and the smaller, 1/4" thick frame that is attached that the structural frame. The acrylic hinges & the thumb tabs that keep the door shut should also be visible.
Sorry for the condition of the cage in this picture. It spent a day under water in a flood (no animals inside, fortunately) and this picture was taken the morning after a Corucia zebrata and her newborn had spent the night trashing the cage.
