Cook the bacon in a large saute pan over medium heat until brown and crispy on both sides. Transfer to a plate lined with paper towels and reserve.
Preheat a griddle or nonstick saute pan to medium. Between two pieces of paper towels, pat dry the mozzarella.
Spread each piece of bread with a thin, even layer of butter. Spread the other side of each piece of bread with Dijon. Lay the bread down with the Dijon sides facing up on a baking sheet.
Place one slice of fontina, one slice of mozzarella and then another slice of fontina on each piece of bread. Lay 3 bacon slices on half of the slices of the cheese-lined bread. Top the bacon with an even layer of apple slices (these are the bottoms, the ones with just cheese are the tops--together they make a pair!).
Place the tops and bottoms on the griddle butter-side down and cook until the bread gets brown and crusty and the cheese begins to melt, 5 to 7 minutes. When the cheese is nice and melty, flip the tops onto the bottoms so the cheese is on the inside. Give each sandwich a press, flip over and cook for 1 to 2 minutes more.
Cut the sandwiches in half on the bias and serve hot.