In a pie plate, whisk together 1/2 cup flour, 1 teaspoon salt, and 1/2 teaspoon pepper. The eggs can be whisked in a separate, shallow bowl or plate.
Chicken breasts should be halved lengthwise, then the halves should be pounded to a thickness of about 1/4 inch using the flat side of a meat mallet sandwiched between two sheets of parchment paper.
Using the same method, dredge the remaining chicken in flour, coating both sides and shaking off any excess before dipping them in the egg mixture. Heat the oil in a large (12-inch) skillet over medium heat until it shimmers.
Put three or four chicken breasts in the pan (or as many as you can fit). Prepare for about 3 minutes per side, or until lightly browned. Cooked chicken should be placed on a plate lined with paper towels before proceeding with the rest of the chicken.
Take out any excess fat before adding the lemon slices to the pan. It takes about 30 seconds per side to get them nice and charred and fragrant. Transfer them to a plate.
Pour in the ½ cup white wine to deglaze the pan, scraping up any browned bits from the bottom. Let it simmer for 2–3 minutes, or until reduced by about half.
Put in the butter and let it soften. Add the flour and cook it for 30 seconds to get rid of the flour flavor. Simmer for 2 minutes, then add the broth, lemon juice, the remaining 1/2 teaspoon salt, and pepper.
The chicken should be cooked through after another 3–4 minutes in the skillet.
Serve with a sprinkle of chopped parsley on top.