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

Melt 1 tablespoon of butter in now empty pot over medium-high heat. Add bell pepper and onion and cook, stirring occasionally, until softened, for about 5 minutes. Transfer to a bowl; set aside. In now-empty pot, melt remaining 2 tablespoons butter over medium heat.

Add flour and cook, whisking constantly, for about 1 minute. Slowly whisk in broth and half-and-half and bring to a boil. Reduce heat to medium-low and simmer until slightly thickened, for about 5 minutes.

Add chicken and cook until no longer pink – 8 to 10 minutes.

Preheat oven to 425 degrees. Remove pot from the heat. Once the chicken is cool enough to handle, transfer chicken to a plate, and shred into bite-size pieces. Whisk cheddar cheese into sauce until smooth. Stir shredded chicken, noodles, bell pepper mixture, mushrooms, 1 1/2 teaspoons salt, and 1 1/4 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.