Heat 1/4 inch of vegetable oil in each of 2 large skillets until shimmering. Sprinkle the fish with more of the flour mixture and add the fillets to the skillets, skin side up. Cook over moderately high heat, turning once, until golden and cooked through, 5 to 6 minutes. Drain the fish on a paper towel–lined baking sheet. Serve right away, with the tartar sauce.