Slice the chicken breasts in half horizontally to make 4 equal chicken fillets. Place in a zip-top bag and pound with a meat tenderizer until 1/4-inch thick.
In a shallow dish, whisk together the 1/4 cup flour, salt, and pepper. In another shallow bowl, lightly whisk the eggs with a fork. In a large skillet, heat the olive oil over medium-high heat.
Dredge the chicken in the flour mixture until evenly coated, shaking off any excess flour. Dip in the egg quickly to coat. Place in the skillet and sauté 3 to 4 minutes per side, or until golden brown and cooked through. If needed, divide chicken into 2 batches if the skillet is too small to cook all at once. Transfer chicken to a plate.
In the same skillet, melt the butter over medium heat. Whisk in the remaining 1 tablespoon flour and cook 30 seconds. Add the broth and lemon juice, whisking continuously and scraping up any browned bits. Add the lemon slices. Simmer 5 minutes, or until the sauce has thickened. Season to taste.
Return chicken to the skillet or place chicken on serving plates and spoon sauce over the top. Sprinkle with parsley to serve.
Notes
Storage: Store chicken and sauce in an airtight container in the fridge up to 4 days. Reheat on the stove over medium-low heat for a few minutes or in the microwave until warmed through.