Bring a large pot of water to a boil. Add pasta and cook until just tender, 4 to 6 minutes or according to package directions. Drain and rinse.
Meanwhile, whisk 5 teaspoons flour and broth in a small bowl until smooth. Place the remaining flour in a shallow dish. Season chicken with 1/4 teaspoon salt and pepper and dredge both sides in the flour.
Heat 2 teaspoons oil in a large nonstick skillet over medium heat. Add the chicken and cook until browned and no longer pink in the middle, 2 to 3 minutes per side. Transfer to a plate; cover and keep warm.
Heat the remaining 1 teaspoon oil in the pan over medium-high heat. Add mushrooms and cook, stirring, until they release their juices and begin to brown, about 5 minutes. Transfer to a plate. Add garlic and wine to the pan and cook until reduced by half, 1 to 2 minutes. Stir in the reserved broth-flour mixture, lemon juice and the remaining 1/4 teaspoon salt. Bring to a simmer and cook, stirring, until the sauce is thickened, about 5 minutes.
Stir in parsley, capers, and the reserved mushrooms. Measure out 1/2 cup of the mushroom sauce. Toss the pasta in the pan with the remaining sauce. Serve the pasta topped with the chicken and the reserved sauce.