Place chicken breasts in a large pot; add enough water to cover. Bring to a boil; reduce heat and simmer until an instant-read thermometer inserted into the center reads at least 165 degrees F (74 degrees C), 15 to 20 minutes. Remove chicken with a slotted spoon and cool until easily handled, about 10 minutes. Reserve cooking liquid.
Dice chicken into bite-size pieces.
Layer 1 bag crushed potato chips in the bottom of a 9x13-inch baking dish. Top with diced chicken.
Preheat oven to 350 degrees F (175 degrees C).
Melt butter in a saucepan over medium heat. Add onion, mushrooms, green bell pepper, and red bell pepper; cook and stir until onion is translucent, 5 to 10 minutes.
Stir in flour into onion mixture to make a paste; cook until golden, about 3 minutes. Pour in 2 cups reserved cooking liquid and milk; cook and stir until sauce is smooth and thick, about 5 minutes. Add American cheese, 1 slice at a time, stirring until melted.
Pour sauce over chicken in the baking dish. Top with remaining crushed potato chips.
Bake in the preheated oven until bubbly and golden, about 30 minutes.