From Serious Eats, a classic method for preparing sole filets (or any thin fish filet), it is also very simple. Basically, season the filet, dredge in flour, saute over high heat in browned butter, and finally use the butter for a sauce.
Ingredients
All-purpose flour, for dredging (about 1/2 cup)
1 1/2 pounds (680g) thin boneless, skinless sole or other white-fleshed fish fillets (see note)
Kosher salt
4 tablespoons (60g) homemade or store-bought clarified butter, plus more if needed (see note)
6 tablespoons (90g) unsalted butter, cut into 1-tablespoon pieces
1 lemon, cut into wedges, plus more for serving
1 1/2 tablespoons (1/4 ounce; 8g) flat-leaf parsley leaves and tender stems
Instructions
1. In a wide, shallow bowl or platter, spread flour in an even layer. Line a rimmed baking sheet with parchment paper. Season fish fillets all over with salt. Working with one piece of fish at a time, dredge fillets in flour, shaking off excess. Transfer to prepared baking sheet.
2. Working in two or more batches so that you can comfortably fit the fillets in one layer without crowding, heat 2 tablespoons (30g) clarified butter in a large nonstick skillet over medium-high heat until shimmering. Add fish, presentation side down, and cook until well browned on on first side, 3 to 4 minutes (it's okay if the fillets are mostly cooked through at this point). Using two spatulas, if necessary, carefully flip each fillet and continue cooking until just cooked through and flesh is opaque throughout, 30 seconds to 1 minute. Carefully transfer fish, well-browned side up, to a serving platter or plates and keep warm. Repeat with remaining fillets and clarified butter, using 2 tablespoons of clarified butter per batch.
3. Wipe out skillet. Add unsalted butter to skillet, set over medium-high heat and cook until butter is melted and foamy. Continue to cook, swirling pan, until milk solids in butter turn a deep hazelnut brown, 1 to 2 minutes longer. While butter is browning, squeeze lemon juice all over fish, then sprinkle all over with parsley.
4. When butter is browned, immediately spoon it all over the fish; it should bubble and foam as soon as it hits the fish (this is very fun to do tableside for your guests and adds to the experience). Serve immediately with additional lemon on the side.
Notes
True sole from the Soleidae family, most commonly Dover sole, can be both hard to find and expensive. Many other thin, white-fleshed flatfish fillets can work in this recipe, including many that are sold as "sole" even if they're not technically in the Soleidae family, including grey sole, Petrale sole, lemon sole, and flounder. The number of fillets you get will depend on the specific fish you buy; 6-ounce (170g) fillets will yield four perfect portions, but some flatfish yield smaller fillets, in which case you may need to divide the fillets up among diners accordingly.
Clarified butter is perfect for browning the fish without the risk of the milk solids found in whole butter burning or scorching, while still reinforcing the buttery flavor of the dish. If you don't have clarified butter, you can brown the fish in a neutral oil like vegetable or canola oil instead.