Place the bacon in a large skillet and cook over medium-high heat, turning occasionally, until evenly browned, about 10 minutes. Drain the bacon slices on paper towels. Chop bacon when cool enough to handle.
Bring a saucepan of water to a boil, reduce heat to medium, and gently lower eggs into the boiling water using a spoon. Boil for 8 minutes. Remove immediately from heat, drain, and cover eggs with cold water.
Heat vegetable oil in a skillet over medium heat; cook and stir chicken cubes, onion, and mushrooms in the hot oil until the meat is no longer pink inside and mushrooms are softened, about 10 minutes.
Place iceberg lettuce into a salad bowl. Add chicken-mushroom mixture and bacon. Top with red bell pepper, cherry tomatoes, and white Cheddar cheese. Stir mayonnaise into salad, tossing gently to coat. Peel eggs and cut each egg in half; cut halves horizontally into 3 wedges apiece. Arrange eggs on top of salad; serve warm.