In a wide shallow bowl, combine the eggs and milk. In another shallow bowl, blend the flour with 1 teaspoon salt and ½ teaspoon pepper. Line a clean counter or a baking sheet with paper towels.
In a large skillet, heat 2 tablespoons of the butter with two tablespoons olive oil until the butter is melted and hot. Lightly dredge each chicken breast in flour, shake off any excess, then dip the chicken in the egg mixture turning to coast. Allow any excess egg to drip back into the bowl. Fry the chicken in batches for about 3 minutes per side, until golden brown and almost cooked through. As the chicken cooks, transfer it to the paper towel-lined surface. Use up to another 2 tablespoons of oil and 2 tablespoons of butter as needed.
When all of the cutlets are browned, pour out any excess fat. Add the lemon slices and cook for 1 to 2 minutes on each side until the lemon slices are slightly browned. Remove the lemons to a plate. Return the pan to medium heat, add a tablespoon more butter, and when the butter is melted, add a tablespoon of the flour you used to bread the chicken and whisk it into the butter. Add the wine and stir to loosen any browned bits from the bottom of the pan. Add the broth and bring to a simmer. Simmer until the sauce has thickened slightly, then return the chicken to the pan (you can overlap the pieces slightly). Cook until the chicken has cooked all the way through and it’s hot, about 3 more minutes.
Serve the chicken with the lemon slices on top, sprinkled with a bit of parsley.
Notes
If you buy thin-sliced cutlets, you are all set to make this dish. If you buy regular chicken breasts, you can pound or slice them to a 1/3-inch thickness. To pound the cutlets, place the chicken breasts in between two pieces of plastic wrap or parchment paper, and use a rolling pin or another sturdy heavy tool to gently pound the chicken until it is of uniform thinness. If the chicken becomes too large and unwieldy, you can cut the breasts in half.
Or you can slice the chicken breasts horizontally to create thinner pieces. Depending on the thickness of the chicken breasts, you may want to slice them into two or three slices horizontally. You want slices ⅓-inch thick.
You will likely want to cook the chicken in batches during the initial cooking, so they have enough room in the pan.
It's fine to overlap the chicken pieces slightly in the pan when you finish cooking them in the sauce.
Allowing the lemons to caramelize in the pan gives them some sweetness. You can eat them rind and all if you like! A little bit of lemon with bites of chicken provides a deep lemon flavor.