Bring a large pot of salted water to a rolling boil and add bowtie pasta. Cook according to package instructions until al dente, about 10 minutes. In the last 3 minutes of pasta cooking, add the broccoli florets. Reserve ½ cup of the starchy pasta water, drain the pasta and broccoli together, and set aside.

Season the chicken breasts liberally with Cajun seasoning. Heat olive oil in a large skillet over medium-high heat. Cook chicken for 6-7 minutes until golden brown, flipping halfway through, until reaching an internal temperature of 165°F.

In the same skillet, melt unsalted butter, then add minced garlic and sauté until fragrant. Stir in crushed red pepper flakes, Dijon mustard, lemon juice, and zest. Simmer briefly, then add chopped parsley and chives; season with salt and pepper.

Return the cooked bowtie pasta, broccoli, and chicken to the skillet with the sauce. Toss together over low heat, gradually adding reserved pasta water to achieve desired sauce consistency. Stir in freshly grated Parmesan cheese until creamy.

Transfer to serving plates, garnishing with additional Parmesan and a squeeze of lemon juice. Serve warm.