Heat a large skillet over medium-high heat. Add oil; swirl to coat. Add garlic to pan; sauté 1 minute. Add spinach and arugula; stir until wilted. Remove spinach mixture from pan. Return pan to medium-high heat. Add bacon; cook until crisp. Remove bacon.
Top 4 bread slices evenly with cheeses, bacon pieces, spinach mixture, and the remaining 4 bread slices.
Heat pan over medium heat. Add sandwiches to drippings in pan; weigh down with a plate. Cook 2 minutes on each side.