Beat eggs in a shallow dish or pie plate. In a large resealable bag, add the flour and salt together, shaking to combine. Place each chicken breast in the flour mixture and shake until fully coated. Dip each chicken breast into the eggs and shake in the flour again.
In a large skillet over medium-high heat and add the oil, and melt 1 tablespoon of butter. Add the chicken and cook for 5 minutes, until browned. Reduce the heat to medium and flip the chicken, cooking 3-5 minutes. Remove chicken from skillet and place on a serving dish; keep warm until ready to serve.
Reduce heat to medium-low, add the garlic, and cook for about 1 minute, until fragrant. Add the broth, wine, lemon juice, and 2 tablespoons of flour into the skillet and gently stir to dissolve the brown bits on the bottom. Simmer for 1 minute, then add in the capers and the remaining butter. Continue to stir while simmering for 5 minutes, until the sauce is thick and glossy. Pour sauce evenly over each piece of chicken, garnish with parsley and serve.