Brush one side of each bread slice with 1/2 tsp. olive oil.
Lay half the slices, oiled side down, on a clean work surface and top with the cheddar cheese, dividing evenly. Place avocado slices on top of the cheddar and the bacon slices next then top with the Monterey jack, dividing evenly.
Top each with one of the remaining bread slices, oiled side up.
Heat a large fry pan or griddle over medium heat until hot. Place 2 of the sandwiches in the pan, reduce the heat to medium-low and cook until the undersides are golden brown, 3 to 4 minutes. Flip the sandwiches and brown the other side, 3 to 4 minutes more. Repeat with the remaining sandwiches.
Transfer the sandwiches to a cutting board and cut in half. Serve immediately. Serves 4.