Prep the lemon. Slice the lemon in half, juice one half, then cut the other half into ⅛" slices and set aside.
Trim and season the chicken. Trim any excess fat from the chicken breasts and slice in half lengthwise to make two thin cutlets. Season both sides of the chicken breasts evenly with the kosher salt and freshly ground black pepper then dredge each breast in the flour, shaking off any excess.
Pan fry the chicken. Heat 2 tablespoons butter with the canola oil in a large skillet over medium-high heat. Add 4 pieces of the chicken and cook for 2-3 minutes per side. You can keep the cooked chicken breasts warm while the sauce cooks by plating on a platter and covering with aluminum foil, or place in a 200°F oven. Continue with the remaining chicken.
Make the piccata sauce. Reduce the heat to medium and add the chicken broth or wine (or ½ cup of both), the lemon juice, sliced lemons, and the capers, scraping up the browned bits on the pan and cook for 2-3 minutes.
Garnish and serve. Stir in the remaining 1 tablespoon of butter until melted. Taste for seasoning and spoon the sauce over the chicken breasts. Top with parsley, if desired. Serve with mashed potatoes or cauliflower, polenta, or buttered noodles.
Notes
Storage: If you end up with leftover lemon chicken piccata, refrigerate it in an airtight container for up to 5 days. Reheat the chicken in a lightly oiled skillet over medium heat, taking care not to overcook it.