Line a baking sheet with aluminum foil. Slice the chicken breasts in half lengthwise. In a shallow bowl whisk together eggs and milk. Set aside. Combine the salt, pepper, and flour in a separate bowl. Dip the chicken breasts one at a time first into the flour, then dip into the egg and then dredge in the flour mixture. Place onto a plate.
In a large skillet heat 2 tablespoons of olive oil and 2 tablespoons of butter over medium heat. Place the chicken breasts into the skillet, as many as will fit at once, probably 2-3 at a time. Cook the chicken until well browned, about 3 minutes per side. Transfer to the baking sheet. Repeat with the remaining chicken breasts adding more olive oil and butter if needed.
Empty the skillet and add 3 tablespoons of butter over medium heat. Add the minced onion and cook, stirring occasionally until sot. Add the wine, lemon juice, and chicken broth. Bring to a boil stirring constantly. Reduce heat to medium and cook the sauce, stirring frequently until the sauce has reduced to about 1 1/2 cups, about 10 minutes. Add the remaining tablespoons of butter to the sauce. Stir until melted.
Add the chicken cutlets to the sauce and heat for 4-5 minutes or until fully cooked (the internal temperature has reached 165 degrees F.).
Serve with pasta and garnish with parsley and lemon wedges for serving.