Thinly slice the chicken breasts lengthwise to create cutlets.
In one shallow bowl, mix flour, salt, and pepper. In another, whisk eggs and milk.
Dredge each chicken cutlet in the flour mixture, then dip in the egg mixture.
Heat olive oil and 2 tablespoons butter in a skillet over medium heat; cook chicken until golden and cooked through, about 3-4 minutes per side. Remove and keep warm.
In the same pan, melt remaining butter, add lemon slices, and sprinkle in flour; cook for 1-2 minutes.
Slowly whisk in white wine and chicken broth, scraping browned bits from the pan. Add lemon juice, salt, pepper, and simmer until slightly thickened.
Return chicken to pan and coat with sauce; sprinkle with chopped parsley before serving.