Cook pasta and broccoli. Boil bowtie pasta in salted water. During the last 3 minutes of cooking, add the broccoli florets to the pot. Reserve 0.5 cup of pasta water, then drain.
Sear the chicken. Heat olive oil in a large skillet over medium-high heat. Season chicken with 0.5 tsp salt and pepper. Cook for 4 minutes, flip, and cook 3-4 more minutes until done. Remove chicken from skillet.
Make the cowboy butter. Lower heat to medium-low. Melt the butter in the same skillet. Whisk in the garlic, lemon zest, lemon juice, parsley, chives, thyme, Dijon mustard, red pepper flakes, and remaining 0.5 tsp salt. Simmer for 1 minute.
Combine everything. Add the cooked chicken, drained pasta, and broccoli into the butter sauce. Toss to coat. Add a splash of reserved pasta water if needed to gloss the sauce.
Garnish and serve. Remove from heat, sprinkle with parmesan cheese, and serve immediately.
