Drain, then rinse and pat dry. Season with 1/4 teaspoon salt and 1/8 teaspoon pepper.
Remove rosemary leaves from sprigs, leaving a small bunch at top of sprig. Thread livers (1 if large, 2 if smaller) onto rosemary sprigs.
Heat 1 tablespoon butter and oil in a 12-inch heavy skillet over medium heat until foam subsides. Cook livers on skewers, turning once, until golden and just cooked through, 5 to 6 minutes total.
Transfer to a platter.
Pour off fat from skillet, then stir remaining tablespoon butter, radishes, and lemon juice into skillet. Spoon over livers.
