Heat a medium pan over medium to medium-high heat. Add extra-virgin olive oil and chicken and season with salt and pepper. Sauté a couple of minutes, then add onion and cook another 5 to 7 minutes until onions are tender and chicken is cooked through. Turn off heat and reserve.
To boiling pasta water, add pasta and salt to season the cooking water. Cook 5 minutes, then add broccoli and cook 3 minutes more or until pasta is cooked to al dente and florets are just tender.
While pasta cooks, heat a medium sauce pot over medium heat. Add butter and melt, then add flour, cayenne and paprika and whisk together over heat until roux bubbles then cook a minute more. Whisk in milk and stock and raise heat a little to bring sauce to a quick boil. Simmer sauce to thicken about 5 minutes.
Drain macaroni or pasta and broccoli florets. Add back to pot and add chicken to the pasta and broccoli. Add cheese to milk sauce and stir to melt it in, about a minute or so. Stir in mustard and season sauce with salt and pepper. Pour sauce over chicken and broccoli and cooked pasta and stir to combine. Adjust seasonings, transfer to a large serving platter and serve.