Prep the salmon. Dry salmon by patting gently on all sides with a paper towel. Combine ½ cup all-purpose flour, 1 tsp Kosher salt, and ¼ tsp black pepper in a shallow bowl. Season salmon filets on all sides with flour, then shake off any excess.
Pan sear the salmon. Heat 1 Tbsp oil and 1 Tbsp butter in a large skillet over medium-high heat. When hot, add salmon, flesh side down, and cook for 3 minutes undisturbed. Carefully turn salmon over so it is now skin side down, then cook an additional 2-3 minutes for medium-rare. Transfer salmon to a plate while you make the sauce.
Make the piccata sauce. Add smashed garlic cloves to the pan, then cook until fragrant and browned, about 30 seconds. Add ½ cup white wine and 2 Tbsp capers, bring to a boil, and cook for about 1-2 minutes to reduce.
Finish the sauce. Pour in ¼ cup stock (or water), turn heat down to medium, then add remaining 4 Tbsp butter. Swirl until butter has completely melted. Remove pan from the heat, add 3 Tbsp fresh lemon juice and2 Tbsp chopped parsley, then stir one final time. Taste and adjust seasoning as necessary.
Finish, then serve. 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.