We are dealing with two different measurements here, frame width and frame thickness. I'm not entirely sure which you are referring to, so I'll cover both. Also, you can reference this useful page: Glossary of Cabinet Hinge Terms
An Overlay door will cover over (lay over) part of the face frame of the cabinet opening. As opposed to an Inset, or Inlay, door which goes inside the opening, flush with the face:
A "Full" Overlay normally means it overlays 3/4" inch. This is because it is made to fully overlap a 3/4" faceless plywood end. These will often be intended for side mounted hinges.
Face Frame Width
A Full overlay would work for you, but you can pick any specific overlay measurement you like where X in a "X" Overlay means the amount it overlaps. 1 1/4" is common for cabinets that DO have face frames (sounds like you do?). You will want frame mounted hinges, preferably (see below).
Face Frame Thickness
However, the next thing you need to consider is how thick the material you used for the face frame is. Common thickness in the USA is 3/4" material. Most frame mount hinges are made to mount on 3/4" unless specified differently.
A good European-style concealed hinge will be adjustable 3 ways. Meaning you can adjust it in/out, up/down, and left/right. As long as your frame is 5/8" to 13/16" (in my experience) it can be adjusted to fit decently.
If your face frame is thinner or thicker than that, you may need to find specialty hinges, or use a side bracket mounted hinge that goes around the face-frame such as these. You will just have to add blocking to the inside to bring the mount out flush with the inside of the frame:
This will work regardless the thickness of your face frame. They could be 2x4 for all they care!
Big thanks to the resources at Rockler Woodworking and Hardware