For the clarified butter: Put butter in a small heavy saucepan over low heat. Do not stir or allow butter to boil. When butter has melted, skim off foam completely and carefully pour the clear yellow fat into a clean container, discarding milky sediment on the bottom of the pan. Store covered in the refrigerator.
For the sole: Pat filets dry with paper towels, season to taste with salt and pepper, then dredge in flour, shaking off any excess. Melt 1⁄2 cup of the clarified butter in each of two large skillets set over medium-high heat. When butter is just smoking, about 2 minutes, place 2 filets in each skillet and cook, basting continually with butter, until lightly browned, about 2 minutes on each side. Add 2 tbsp. of the regular butter to each skillet, season to taste with salt and pepper, and baste filets for 1 more minute, then transfer with a slotted spatula to a serving platter.
Melt the remaining 6 tbsp. of regular butter in a small skillet over high heat. Swirl pan over heat until butter foams and turns light golden, about 1 minute. Whisk in lemon juice and pour over fish filets. Serve fish garnished with lemon sections and fresh herbs, if you like.