Cooking the egg noodles until just al dente and then shocking them in cold water prevents them from overcooking.

Bring 4 quarts water to boil in Dutch oven. Add noodles and 1 tablespoon salt and cook, stirring often, until just al dente, about 3 minutes. Drain noodles and rinse with cold water until cool, about 2 minutes. Drain again and set aside.

Melt 1 tablespoon butter in now empty pot over medium-high heat. Add bell pepper and onion and cook, stirring occasionally, until softened, about 5 minutes. Transfer to bowl; set aside. In now-empty pot, melt remaining 2 tablespoons butter over medium heat. Add flour and cook, whisking constantly, for 1 minute. Slowly whisk in half-and-half and broth and bring to boil. Reduce heat to medium-low and simmer until slightly thickened, about 5 minutes. Add chicken and cook until temperature registers 160 degrees, 8 to 10 minutes.

Adjust oven rack to upper-middle position and heat oven to 425 degrees. Remove pot from heat, transfer chicken to plate, and shred into bite-size pieces once cool enough to handle. Whisk American and cheddar cheeses into sauce until smooth. Stir shredded chicken, noodles, bell pepper mixture, peas, 1 ½ teaspoons salt, and 1 ¼ teaspoons pepper into cheese sauce.

Transfer mixture to 13 by 9-inch baking dish and top with crackers. Bake until golden brown and bubbling, about 15 minutes. Let casserole cool on wire rack for 10 minutes. Serve.