1 small head broccoli, cut into small florets (about 4 cups)
8 oz. extra-sharp cheddar cheese, grated (about 2 cups)
Sliced fresh chives, for garnish
Cook pasta according to package directions; set aside.
Season chicken with salt and pepper. Melt 1 tablespoon butter in a large pot or Dutch oven over medium-high heat. Add chicken and cook, stirring occasionally, until golden and cooked through, 6 to 9 minutes. Remove chicken to a plate.
Reduce heat to medium. Add onion, bell pepper, and remaining 3 tablespoons butter to pot. Cook, stirring occasionally, until softened, 3 to 5 minutes. Add flour and cook, stirring constantly, 1 minute. Slowly stir in stock, then milk and mustard. Simmer until thickened, about 7 to 8 minutes.
Stir in broccoli and reserved chicken. Simmer until broccoli is tender and soup is thickened, 4 to 6 minutes. Remove from heat and let stand 1 minute (this prevents soup from breaking). Stir in cooked pasta. Gradually stir in cheese until melted and smooth. Season with salt and pepper. Garnish with chives.