Combine flour, Kosher salt, and black pepper in a shallow bowl, then mix well. Dredge each chicken cutlet in the seasoned flour, making sure to coat evenly on all sides. Shake off excess flour, then transfer to a plate.
Heat olive oil and butter in a large skillet over medium-high heat. Add the cutlets, then cook undisturbed until golden brown. Add olive oil and butter, if needed, when cooking the second batch.
To the pan, add shallot and lemon. Season lightly with Kosher salt, then cook until fragrant and softened. Add the smashed garlic cloves, then cook. Pour in white wine, then use a spatula to scrape any browned bits from the bottom of the pan. Cook.
Remove the pan from the heat, then swirl or whisk in the remaining butter until melted. Taste and adjust seasoning. Serve the chicken with the sauce poured on top, then garnish with chopped parsley and a pinch of flaky sea salt.