Cook the bacon in a large skillet over medium heat until crisp, 6 to 8 minutes. Let cool on a paper towel-lined plate. Break into pieces. Add the oil to the drippings in the skillet.
Season the chicken with ¼ teaspoon each salt and pepper and cook until cooked through, 2 to 4 minutes per side; slice.
Toss the romaine with the Cheddar, red onion, vinaigrette, bacon, and chicken in a large bowl.