Bring a large pot of salted water to a boil. Cook the pasta according to package directions until al dente. In the last 3 to 4 minutes, add the broccoli florets and cook until crisp-tender. Reserve 2 cups of pasta water, then drain the pasta and broccoli. Set aside.
In a small bowl, combine salt, pepper, dried mustard, and paprika. Season chicken evenly with the seasoning mixture.
To a large nonstick skillet over medium heat, add oil. Once hot, add chicken breasts. Cook chicken for 5 minutes. Then, flip over the chicken, reduce heat to medium-low, and cook for an additional 5 minutes, or until the chicken reaches an internal temperature of 165°F. Remove from heat. Transfer the chicken to a plate. Tent to keep warm.
In the same skillet over medium heat, melt the butter. Add the garlic and cook for 1 minute.
Whisk in flour and cook for 1 minute to form a roux.
Slowly whisk in milk, scraping up any browned bits from the pan. Stir in dry mustard, salt, and red pepper flakes (if using).
Reduce heat to low and simmer for 3 to 4 minutes until the sauce thickens slightly and coats the back of a spoon.
Add the shredded white cheddar a handful at a time, stirring until fully melted and the sauce is smooth and velvety. If the sauce is too thick, stir in a splash of the reserved pasta water until it reaches your desired consistency.
Add pasta and broccoli to the sauce and toss to coat.
Slice or chop chicken and add back in (or stir through). Heat gently until everything is combined.
Garnish with fresh parsley and serve warm.