Preheat the oven to the lowest setting. Add a platter and 4 dinner plates; let warm while you make the tilapia.
Pat the fish dry; season with salt and pepper. Place the flour on a large plate. Dredge the fish lightly in the flour, shaking off the excess.
In a large skillet, heat 2 tbsp. EVOO, two turns of the pan, over medium-high. When the oil begins to smoke, melt in 1 tbsp. butter. Add 2 fillets and cook until opaque in the center, 2 to 4 minutes per side. Transfer the fish to the warm platter in the oven. Wipe the skillet clean, then repeat with the remaining 2 tbsp. EVOO, 1 tbsp. butter and 2 fillets.
Wipe the skillet clean, then add 1 tbsp. butter. When the butter melts, add the lemon slices and cook, turning once, until lightly browned, 2 to 3 minutes. Place the lemon slices on top of the fish.
Add the shallot and garlic to the skillet and stir until softened, about 1 minute. Add the wine, capers, parsley and lemon juice. Swirl in the remaining 1 tbsp. butter and simmer until the sauce thickens slightly, about 1 minute. Divide the fish among the warm plates. Season the lemon sauce and spoon over the fish.