Bring 4 quarts of water to a boil in a Dutch oven.

Add the noodles and 1 tablespoon salt, cooking while stirring frequently until just al dente, about 3 minutes.

Drain the noodles and rinse under cold water until cool, about 2 minutes. Drain again and set aside.

Melt 1 tablespoon butter in the now-empty pot over medium-high heat.

Add the bell pepper and onion and cook, stirring occasionally, until softened, about 5 minutes.

Transfer to a bowl and set aside.

In the same pot, melt the remaining 2 tablespoons butter over medium heat.

Add the flour and cook, whisking constantly, for 1 minute.

Gradually whisk in the half-and-half and broth, then bring to a boil.

Reduce the heat to medium-low and simmer until the mixture thickens slightly, about 5 minutes.

Add the chicken and cook until it reaches an internal temperature of 160°F, about 8 to 10 minutes.

Adjust the oven rack to the upper-middle position and preheat the oven to 425°F.

Remove the pot from heat, transfer the chicken to a plate, and shred into bite-sized pieces once cool enough to handle.

Whisk in the American and cheddar cheeses until smooth.

Stir in the shredded chicken, noodles, bell pepper mixture, peas, 1 teaspoon salt, and 1 teaspoon pepper.

Transfer the mixture to a 13×9-inch baking dish and top with crushed crackers.

Bake until golden brown and bubbling, about 15 minutes.

Let cool on a wire rack for 10 minutes before serving.