Pound cutlets to 1/4 inch thick; season with salt and pepper. Sprinkle with juice of 1/2 lemon, turning to evenly coat. Stir together Parmesan and 1/3 cup all-purpose flour on a plate. Dredge cutlets in cheese mixture, pressing to adhere.

Heat 2 tablespoons olive oil in a large nonstick skillet over medium-high. When oil shimmers, add 4 cutlets in a single layer; cook, flipping once, until golden brown and crisp, 4 to 5 minutes. Transfer to a plate. Wipe skillet clean; repeat with 2 more tablespoons oil and remaining 4 cutlets. Wipe skillet clean; heat 2 tablespoons oil in skillet over medium-high. Add polenta and cook, flipping once, until golden and crisp, about 5 minutes.

Meanwhile, cook asparagus in a pot of generously salted boiling water until crisp-tender, 2 to 3 minutes. Drain and transfer to plates. Season with salt and pepper, drizzle with oil, and top with finely grated lemon zest and shaved Parmesan. Serve with cutlets, polenta, and lemon wedges.