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 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 a bowl set aside. Melt remaining 2 Tablespoons butter in again-empty pot over medium heat. Add flour and cook, whisking constantly, for 1 minute. Slowly whisk in half-and-half and broth, scraping up any browned bits and smoothing out any lumps, and bring to boil. Reduce heat to medium-low and simmer until slightly thickened, about 5 minutes. Add chicken and cook until no longer pink, 8 to 10 minutes.
Adjust oven rack to upper-middle position and heat oven to 425 degrees. Off heat, transfer chicken to plate. Let chicken cool slightly, then shred with 2 forks into bite-size pieces. Whisk cheese into sauce until smooth. Stir peas, pepper, salt, noodles, bell pepper mixture, and shredded chicken into cheese sauce.
Transfer mixture to 13 by 9-inch baking dish (or similar size casserole dish) and top with crackers. Bake until golden brown and bubbling, about 15 minutes. Let casserole cool on wire rack for 10 minutes before serving.