Maybe try a piano hinge?

That won't work since there is an overlay - maybe this? http://www.amazon.com/Blum-Compact-Soft-Close-BLUMotion-Overlay/dp/B007O6VFMA

I don't think this is what you want and I'm not sure what they are called, but here's what we have.

Yes, either a standard butt type hinge as crl posted, or a piano hinge is what you need.

Is there any reason we can't use a hinge like store bought cabinets come with? We have access to a drill press so I don't think installing it will be that big of a deal. We were hoping to have it match the rest of our cabinets without any exposed hinges.

There are multitudes of hinges and they each require specific ways to be mounted. Are you asking about using a European style hidden hinge? You can get a templet to bore the hinge recess into the door, but there needs to be someplace on the cabinet box frame to attach the hinge. If you are near a Woodcrafters retail store take the door and a photo of your cabinet box to them and they can help you.

Easiest would be an surface mount hinge, like this

Next would be a semi-concealed wrap around hinge, since you do not need to cut a hinge cup. like this

If you use a compact type concealed hinge, you'll probably have to cut a small notch in the wall/frame for the legs on the cabinet side of the hinge to sit.

You could probably use a euro type concealed hinge.

This is all assuming you don't have a door for it yet.

Show a picture of the top door. My guess is that it's a 3/8ths inset / partial inset

The plywood sides you added are going to be in the way of a European hinge working. If you shop online for concealed hinges, every product is going to have a diagram showing the relationship of the door to the cabinet box or faceframe. Not ever one is capable of envisioning the relationships in their head.

Using the original hinge is going to save a lot of time and trial & error until you figure it out (or realize it's impossible to do).

Might help to see the door, see the edge detail, see how it fits the opening.

Casey