Pat the sole fillets completely dry with paper towels. Season both sides with salt and pepper.
Spread the flour in a shallow dish. Dredge each fillet in flour, shaking off the excess, and set on a plate.
Heat 1 tbsp olive oil and 1.5 tbsp butter in a 12-inch skillet over medium-high heat until the butter foams.
Add 2 fillets to the skillet, don't crowd the pan, and cook 2 to 3 minutes per side until golden and the fish is opaque and flakes easily, about 145 F / 63 C. Transfer to a warm plate.
Wipe the skillet clean, add the remaining 1 tbsp olive oil and 1.5 tbsp butter, and repeat with the remaining 2 fillets.
Wipe the skillet clean again. Add the remaining 4 tbsp butter over medium heat and swirl, cooking 3 to 4 minutes until it turns amber and smells nutty, watching closely so it doesn't burn.
Remove the skillet from heat and stir in the capers and lemon juice.
Spoon the brown butter caper sauce over the fillets, sprinkle with parsley, and serve with lemon wedges.