Whisk together flour, black pepper, and paprika in a shallow dish. Dredge the chicken pieces in the flour mixture, coating evenly; set aside. Heat olive oil in a large skillet over medium-high heat; stir and cook the garlic until light brown, about 1 minute. Remove the garlic from the skillet and set aside.
Place 2 tablespoons of butter into the skillet with the olive oil. Cook the chicken pieces in the oil and butter over medium-high heat until brown, about 5 minutes per side. Remove the chicken from the pan and set aside.
Pour the wine into the hot skillet and bring to a boil over high heat, scraping the browned pieces from the bottom and sides of the pan. Boil the wine until it is reduced by half, about 5 minutes. Whisk in the chicken broth, reserved garlic, lemon juice and capers. Cook for 5 minutes over medium-high heat. Stir in the remaining 2 tablespoons butter and parsley. Return the chicken pieces to the skillet and continue cooking over medium heat until the sauce thickens, about 15 minutes.
Meanwhile, fill a large pot with lightly salted water and bring to a rolling boil over high heat. Once the water is boiling, stir in the angel hair pasta, and return to a boil. Cook the pasta uncovered, stirring occasionally, until the pasta has cooked through, but is still firm to the bite, 4 to 5 minutes. Drain well.
Remove the chicken pieces to a serving dish and drizzle with a few tablespoons of the sauce and capers. Place the cooked angel hair pasta into the skillet with the remaining piccata sauce and toss to coat.