In a grill pan over medium-high heat, melt 2 tablespoons of the butter. Cook the apples until just softened, about 5 minutes. Remove the apples from the pan and reserve; don't clean the pan.
Top each of the 4 bread bottoms with one-fourth of the cheese, turkey and cooked apples, and the bread tops.
Melt 1/2 tablespoon of butter in the grill pan over medium heat. Add 1 sandwich and place a heavy pan on top of it to press it down. Grill for 3 minutes, pressing down occasionally, then turn over and continue cooking until the bread is toasted and the cheese has melted, about 2 minutes more. Grill the remaining sandwiches, adding another 1/2 tablespoon of butter each time before grilling.