Doors – How to install a garden door that’s thicker than the wall

Looks like they do sell pet doors for that purpose. You can even pick the size.

As far as installing it, I think the last thing you want to do is center it on a stud.

From you description, it sounds like you have a hair over 1 1/2 inches of extra space to make up. If this is correct consider this method. Place a full 1x4 (or whatever the depth of the wall may be) on both sides, as long as the new door frame will still fit comfortably in the new opening. When ever possible, I like to mount the hinge side to a solid petition without excessive shims. This makes it more solid without having to bridge a large gap with your finish nails, and makes it a lot easier to perfectly plumb the hinge jam. I often replace the center hinge screw on the jam side with a longer matching screw that will attach to the jam. This does a few things for you. One, you don't need as many nails, gives a much more solid feeling and sounding door, and gives you a tool to fine adjust the plumb if necessary.(especially on heavier solid wood doors. Just like with security screws on an entry door. After you are sure you are happy with the plumb on the hinge side, simply use visual to get an even gap on the top and latch side, in that order. You can then use shims on the latch side and shoot your finish nail just below your shims. this keeps the shims from dropping as you work down the jam. i don't usually shoot through the shims because they often split and I may want to slightly adjust them before the trim casings are installed.