Bring 4 quarts of water to a boil in a Dutch oven. Add 1 tbsp salt and noodles. Cook until al dente, about 3 mins.
Drain, rinse under cold water, drain again, and set aside.
Melt 1 tbsp butter in the empty pot over medium-high heat. Sauté bell pepper and onion until softened, about 5 mins. Transfer to a bowl.
Melt remaining 2 tbsp butter in pot. Add flour and whisk for 1 min. Gradually whisk in half-and-half and broth, bring to a boil. Reduce heat and simmer until thickened, about 5 mins.
Add chicken, cook until internal temperature reaches 160°F, about 8-10 mins.
Preheat oven to 425°F. Remove chicken, shred when cool.
Whisk American and cheddar cheese into sauce. Stir in chicken, noodles, vegetables, peas, 1 tsp salt, and 1 tsp pepper.
Transfer to a 13x9-inch baking dish. Top with crushed crackers.
Bake until golden brown and bubbling, about 15 mins.
Cool on wire rack for 10 mins before serving.