Boil the pasta in salted water, adding the broccoli during the last 3 minutes, and reserve 1/2 cup of pasta water before draining.
Sear the cubed chicken in a skillet with olive oil and 2 tablespoons of butter until fully cooked, then set aside.
Melt the remaining butter in the skillet, whisking in the garlic, mustard, lemon juice, zest, and spices until fragrant.
Whisk the reserved pasta water into the butter mixture vigorously to create a glossy emulsion.
Toss the chicken, pasta, and broccoli in the sauce, garnishing with parsley and Parmesan before serving.