Bring a large pot of salted water to a boil and add the bowtie pasta. Cook until al dente according to package directions. In the last 2–3 minutes of cooking, add the broccoli florets. Reserve 1/2 cup of pasta water, then drain and set aside.
Season the chicken with Cajun seasoning, garlic powder, salt, and pepper. Heat the olive oil in a large deep skillet over medium-high heat, add the chicken, and cook until browned and cooked through. Transfer the chicken to a plate.
Reduce the heat to medium and add the butter to the same skillet. Once melted, stir in the minced garlic and cook for 1–2 minutes until fragrant. Add the red pepper flakes if using.
Whisk in the Dijon mustard, lemon juice, and lemon zest, then pour in the chicken broth. Let the mixture simmer for a few minutes until slightly reduced and flavorful.
Stir in the heavy cream and Parmesan cheese until the cowboy butter lemon sauce is smooth and lightly thickened.
Add the cooked pasta, broccoli, and reserved chicken to the skillet. Toss gently, adding splashes of pasta water as needed until everything is coated in the sauce.
Sprinkle in the chopped parsley, taste, and adjust seasoning with more salt, pepper, or lemon juice. Serve warm with extra Parmesan and lemon wedges.