Cook pasta according to package directions; set aside, reserving the pasta water.
Season chicken with salt and pepper. Melt 1 tablespoon butter in a large soup pot or Dutch oven over medium-high heat. Add chicken and cook, stirring occasionally, until golden and almost cooked through, 6 to 7 minutes. Remove chicken to a plate.
Reduce heat to medium. Add the remaining 3 tablespoons butter, onion, and bell pepper. Cook, stirring occasionally, until the vegetables are just starting to soften, about 3 minutes.
Stir in the cauliflower and the broccoli and continue to cook until they are slightly tender, about 5 minutes more.
Add gluten-free flour to coat and cook, stirring constantly, 1 minute.
Slowly stir in stock, then coconut milk, mustard and nutritional yeast. Simmer until thickened and all the vegetables are tender, about 7 to 8 minutes.
Stir in reserved chicken. Simmer gently until the soup is thickened and the chicken is cooked but still tender, 4 to 6 minutes. If it is thicker than you like, add more broth. Season with salt and pepper.
Remove from heat. Stir in cooked pasta. Serve the cheese on the side for anyone who wishes to add it. Garnish with chives.