Sprinkle both sides of cutlets generously with salt and pepper. Dredge cutlets in the flour (I shake them up in a paper bag).
Heat a heavy-bottomed 12-inch skillet over medium-high heat until hot, about 2 minutes; add a tablespoon of oil; swirl pan to coat.
Lay the chicken cutlets in and sauté, without moving them, until lightly browned on first side, 2 to 2½ minutes. Turn cutlets and cook until second side is lightly browned, 2 to 2½ minutes.
Remove cutlets from the skillet to serving plates and let rest while you make the sauce.
Add shallot and garlic to empty skillet and sauté until fragrant. Add the stock/broth and lemon slices, increase heat to high and scrape loose any browned bits.
Simmer until liquid reduces to about 5 or 6 tablespoons, about 4 minutes. Add lemon juice and capers and simmer until sauce reduces again to 5 or 6 tablespoons, about 1 minute.
Remove pan from heat and swirl in butter until butter melts and thickens sauce; swirl in parsley.
Spoon sauce over chicken and serve immediately with a side of pasta and/or vegetables.