Place 4 slices of bread on a cutting board. On each slice, layer cheese, warmed dressing, cranberry sauce, warmed sliced turkey, and warmed mashed potatoes; add another layer of dressing and cheese, then top with the remaining bread.
Preheat a panini press, or heat oil in a grill pan or large skillet over medium-high heat.
Butter the top slice of bread for each sandwich, then carefully transfer the sandwich to the panini press, grill pan, or hot skillet, butter-side down.
If using a skillet or grill pan, optionally weigh the sandwich with a heavy skillet and cook for 2 to 3 minutes. Check the underside: if browned, add butter to the top layer of bread and carefully flip. If not browned, cook another minute or so until browned, then flip. (No need to do this step if using a panini press.)
Remove from the skillet or press, slice, and serve with warmed pan gravy for dipping.