How to Make It
In a large nonstick frying pan, heat 1 tablespoon each of the oil and the butter over moderate heat. Season the chicken with 1/4 teaspoon each of the salt and pepper. Cook the breasts until browned and just done, 4 to 5 minutes per side. Remove the chicken from the pan and let it rest for 5 minutes. Cut into small pieces.
In the same pan, heat the remaining 1 tablespoon oil over moderately low heat. Add the red onion and cook, stirring occasionally, until starting to soften, about 3 minutes. Stir in the garlic, Brussels sprouts, broth, and red-pepper flakes. Bring to a simmer and cook,covered, until the Brussels sprouts are just done, about 5 minutes. Add the chicken, lemon juice, parsley, Parmesan, and the remaining 2 tablespoons butter, 3/4 teaspoon salt, and 1/4 teaspoon pepper. Remove from the heat.
In a large pot of boiling, salted water, cook the pasta until just done, about 10 minutes. Drain and toss with the sauce.