How to Make It
Cook 1/2 cup butter in a medium saucepan over medium-high, stirring often, until browned, about 5 minutes. Stir in sliced garlic; cook 1 minute. Whisk in stock and miso until combined. Bring to a boil; cook until reduced to about 1 cup, 10 to 12 minutes. Transfer to a blender; secure lid, and remove center piece to allow steam to escape. Place a towel over opening. Process until smooth, about 1 minute. Stir in lemon juice and reserve.
Stir together flour and granulated garlic on a plate. Sprinkle fish with salt; dredge fish in seasoned flour. Heat 1 tablespoon oil in a large skillet over high until it begins to smoke. Add lemon slices; cook until just browned, about 1 minute per side. Wipe skillet clean.
Heat 1 tablespoon oil in same skillet over medium-high. Add 2 fillets; cook until golden brown, 4 to 5 minutes per side. During the last minute of cooking, add 11/2 teaspoons butter, 1 garlic clove, and 1 thyme sprig to skillet; spoon mixture over fish. Discard garlic and thyme; transfer fish to a platter and keep warm. Repeat with remaining oil, fish, butter, smashed garlic, and thyme. Add caperberries, capers, and lemon slices to skillet; toss until warmed, about 1 minute. To serve, spoon warm sauce onto 4 plates; top with fish. Spoon caperberries, capers, and lemon over each fillet. Garnish with parsley.