Dry salmon by patting gently on all sides with a paper towel. Combine all-purpose flour, Kosher salt, and black pepper in a shallow bowl. Season salmon filets on all sides with flour, then shake off any excess.
Heat oil and butter in a large skillet over medium-high heat. When hot, add salmon and cook. Carefully turn salmon over so it is now skin side down, then cook. Transfer salmon to a plate while you make the sauce.
Add smashed garlic cloves to the pan, then cook until fragrant and browned. Add white wine and capers, bring to a boil, and cook.
urn heat down to medium-low, then add remaining butter. Swirl until butter has completely melted. Remove the pan from the heat, add fresh lemon juice and chopped parsley, then stir one final time. Taste and adjust seasoning as necessary.
Return cooked salmon filets to piccata sauce, garnish with additional parsley, lemon wedges, and a sprinkle of flaky sea salt. Serve immediately with extra piccata sauce spooned on top.