Bring a large pot of salted water to a boil. Cook bowtie pasta to al dente according to package directions. Add broccoli florets in the last 2 to 3 minutes. Reserve 1/2 cup pasta water before draining. Drain and set aside.

Toss chicken pieces with Cajun seasoning, garlic powder, smoked paprika, salt, and pepper. Heat olive oil in a large skillet over medium-high heat. Add chicken in a single layer, sear undisturbed for 2 minutes, then cook for 6 to 7 minutes total until golden and cooked through. Remove and set aside, leaving browned bits in the pan.

Reduce heat to medium. Add butter to the skillet and melt, scraping up the browned bits. Add minced garlic and cook for about 2 minutes until fragrant. Stir in red pepper flakes, Dijon mustard, lemon juice, and lemon zest. Season with salt and pepper. Stir in parsley and chives.

Return chicken, pasta, and broccoli to the skillet. Toss to coat. Add reserved pasta water a splash at a time until the sauce loosens and clings to the pasta.

Stir in Parmesan until melted and creamy. Serve warm with extra Parmesan and a squeeze of lemon.