Preheat a griddle over medium-high heat.
Lay the bacon strips on the griddle. Cook the bacon until browned and crisp. Drain the bacon on paper towels and reserve the bacon fat.
To build your grilled cheese, place 1/4 cup of the cheese on the bottom of a slice of bread. Top with 2 slices bacon, 4 slices apple, some arugula, 1/4 cup more of the cheese and another slice of bread. Repeat with the remaining ingredients.
Spread each side of the sandwich with some of the butter. Griddle until the bread is toasted, golden brown and delicious, 2 to 3 minutes on each side. Cut each sandwich into 4 pieces and serve.