On a dinner plate, mix flour, cornmeal, salt, and pepper. Rinse trout and pat dry; place in flour mixture and turn to coat.
In a 1-quart pan over medium heat, bring butter to a simmer; remove from heat. With a spoon, skim off and discard foam, leaving clarified butter behind.
Pour 1 tablespoon of clarified butter into a 10- to 12-inch nonstick frying pan over high heat; place trout, skin side down, in pan and cook until browned on the bottom, 2 to 3 minutes. Turn with a wide spatula, reduce heat to medium, and cook until fish is barely opaque but still moist- looking in center of thickest part (cut to test), 2 to 4 minutes longer.
Meanwhile, add capers to remaining clarified butter in pan and shake pan often over medium heat until capers pop open, 1 to 2 minutes.
Transfer trout, skin down, to a plate. Spoon caper butter over fish and garnish with lemon wedges and parsley sprigs. Add salt and pepper to taste.