Place flour in a shallow bowl. Beat eggs in a separate shallow bowl. Coat each piece of chicken in flour; dip in beaten eggs to coat completely. Press chicken into flour again until evenly coated. Reserve remaining flour.
Coat a skillet with olive oil and set over medium heat. Cook each piece of coated chicken in the hot olive oil until browned, about 5 minutes per side. Transfer chicken to a 9x13-inch baking dish.
Preheat oven to 300 degrees F (150 degrees C).
Combine chicken broth, white wine, butter, lemon juice, garlic, parsley, salt, and pepper in a saucepan over medium heat; cook and stir until sauce is smooth, not allowing it to boil, about 5 minutes. Slowly sift 1 cup reserved flour into sauce while continually whisking until sauce thickens, 2 to 3 minutes. Pour sauce over chicken.
Bake chicken in the preheated oven until no longer pink in the center and juices run clear, 20 to 25 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).