1Heat 2 tablespoons each butter and olive oil in a large skillet until shimmering hot. Meanwhile, season four 3-ounce chicken cutlets with salt and pepper, then dust with flour, shaking off any excess.
2Brown the chicken on both sides, 1 to 2 minutes per side. Transfer to a plate. Pour off any fat, add 1/4 cup dry white vermouth to the pan, and deglaze, scraping up any browned bits. Add 1/4 cup chicken stock and cook until syrupy.
3Add 1 thinly sliced, seeded lemon and 2 tablespoons drained capers and cook for a few minutes, until the sauce has thickened slightly.
4Season to taste with pepper and a bit of salt (capers are salty), then swirl in 2 tablespoons unsalted butter and sprinkle with chopped Italian parsley. Return the chicken to the pan for 1 to 2 minutes to reheat, then serve immediately.