Creating a closet where a window used to be

I'm going to be putting a 30" wide x 48" tall closet in my bathroom wall where a window used to be. We had added a garage to that side of the house and removed the window and drywalled over it years ago. There is a cupboard hanging there now, but I think it looks a little too tacked onto the wall and I want to cut away the wall and build a closet that's 18" deep that will hang out into the garage. I plan to build the actual closet of 3/4" plywood, glued and screwed together, then screwing it into the studs in the wall. I don't think any outside (garage side) support will be necessary since it will just hold towels and bathroom supplies. I plan to buy a bifold masonite door and cut it down to 48". I know I'll more than likely have to add some wood to the inside of the bottom since these doors are hollow, but I've done that before on another door, so I don't think that will be a problem. My biggest question is about the best way to insulate it on the garage side. Could I just add furring strips and put rigid foam insulation on it, then put drywall over that (attached to the furring strips)? The garage is unheated and it gets pretty cold in the winter months, so I wondered if the rigid foam would be enough insulation. I would of course caulk all around it before adding the furring strips. What do you think? Just wanted to get some opinions before I knocked a hole in the wall.