Season fish with salt and pepper and squeeze 1 lemon half over fish.
Heat 2 tablespoon butter and oil in a large skillet over medium heat until bubbling.
Spread flour into a shallow bowl. Dredge fish through the flour until lightly coated on both sides.
Cook fish in the bubbling butter mixture, working in batches, until fish is golden on both sides and flakes easily with a fork, 5 to 10 minutes per side. Transfer cooked fish to a serving platter.
Increase heat under skillet to medium-high; add remaining 2 tablespoons butter. Cook and stir butter until fragrant and starting to turn brown, 2 to 3 minutes. Add garlic to brown butter and remove skillet from heat. Stir capers into garlic mixture and squeeze remaining lemon half over mixture. Pour caper sauce over cooked fish.