In a small bowl, stir together the mayonnaise, cranberry sauce, and mustard.
Spread 1 side of each bread slice with the butter. Place 2 slices of bread, buttered sides down, in a large skillet off the heat. Spread the unbuttered side of each slice with a little cranberry mayo. Pile on the turkey, then the stuffing, then the avocado, and then the cheese. Spread a little more cranberry spread on top of the cheese. Add the second slices of bread, buttered side up.
Turn the heat to medium and place the skillet on the burner. Cook, flipping once, until the sandwiches are golden brown, 2 to 3 minutes per side.
Cut the sandwiches in half and serve immediately with the remaining cranberry mayo sauce on the side for dipping.
Notes
Other dip options--If you have leftover gravy, warm some up and use it as a dip along with the cranberry mayo.
Roast turkey substitute--If you don't have roast turkey, this is still great with whatever's in the fridge—leftover pork roast, prime rib, roast chicken, or even canned tuna.
Alternative cooking methods--You can also make these in a panini press or waffle maker, which will allow you to avoid flipping the sandwich.