Heat oil in a large skillet over medium heat; cook the chicken breasts in hot oil until no longer pink in the center and the juices run clear, 13 to 15 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Remove and keep warm.
Place bacon in large skillet and cook over medium-high heat, turning occasionally, until evenly browned, about 10 minutes. Drain bacon slices on paper towels; reserve 2 tablespoons drippings. Cook and stir onion and brown sugar in reserved drippings until onion is golden, about 5 minutes.
Place two bacon strips on each chicken breast half; top with caramelized onions and sprinkle with Colby-Monterey Jack cheese.