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 2 bacon slices on top of the cheddar and 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.