SEASON THE FILLETS with salt and pepper. Combine the fish with the buttermilk and tarragon, in a single layer, in a large saute pan over medium heat. When the buttermilk begins to simmer, cover the pan and cook 2 minutes. Gently roll the fillets over, and cook another minute until the fish is firm but not falling apart.
WITH A SLOTTED SPATULA, remove the fillets to a plate and keep warm, either by covering with foil or by placing in a warm oven. Remove the tarragon sprig and discard. Transfer the buttermilk, which will have separated during cooking, to a blender and add the remaining 4 Tbsp. of butter. (If you have one of those handheld immersion blenders, this is a prime time to use it.) Blend until the butter is smoothly incorporated and the buttermilk is no longer separated. Add the lemon juice, and salt and pepper to taste