Preheat a gas or charcoal grill to high. Shape the ground turkey into 4 round patties about 1 1/2-inches thick. Make a small indentation in the center of each patty, place 1 cube of cheese in it, and press the turkey up and around, to cover it. Brush both sides of the burger lightly with the olive oil and season with salt and pepper.
Grill until cooked through, about 4 minutes on each side.
While the burgers are cooking, brush the apples on both sides with vegetable oil and grill until golden, about 2 minutes on each side.
Split the bread and toast on the grill, cut side down, until golden. Place a burger on the bottom of each quarter. Top with 1 or 2 apple slices, several sprigs of arugula, and the remaining bread. Arrange the burgers on a platter.