1 cup all-purpose flour
3 large eggs, beaten
2 cups panko bread crumbs (4 ounces; 115g), roughly crushed by hand if very large
1/2 cup grated Parmigiano-Reggiano cheese
8 Alaska sole fillets (4-6 ounces each)
Kosher salt and freshly ground black pepper
Vegetable oil, for frying
1/2 cup dry white wine
1 1/2 tablespoons capers, drained
5 tablespoons unsalted butter and a dash of cream (optional)
2 tablespoons fresh juice from 1 lemon
1 1/2 tablespoons minced flat-leaf parsley
1. Set three wide, shallow bowls on a work surface. Add flour to the first one, beaten eggs to the second, and panko and Parmesan cheese to the third. Mix panko and Parmesan thoroughly.
2. Line a rimmed baking dish with parchment paper. Lightly dry sole and season both sides with salt and pepper. Working with one at a time, dredge a fillet in flour with your left hand, shaking off excess. Transfer to egg dish; turn fillet with your right hand to coat both sides. Lift and allow excess egg to drain off; transfer to bread crumb mixture. With your left hand, scoop bread crumbs on top of fish and gently press, turning to ensure a good layer of crumbs on both sides. Transfer to prepared baking sheet and repeat with remaining fillets.
3. Fill a large skillet with 1/4 inch oil. Heat on high until oil shimmers and is just shy of smoking, about 375°F (200°C) on an instant-read thermometer.
4. Working in batches, gently lower fish into pan, laying them down away from you to prevent hot fat from splashing toward you. Fry, gently swirling pan and rotating for even browning, and adjusting heat as necessary for a steady, vigorous bubble, until bottom side is browned and crisp, about three minutes. Flip fish and fry until other side is browned and crisp, about three minutes longer. Transfer to paper towels to drain and season with salt right away. Repeat with remaining cutlets, adding oil if necessary.
5. Drain all but one tablespoon oil from skillet. Add wine and bring to a boil over medium-high heat, cooking until raw alcohol smell has mostly cooked off, about two minutes. Add capers and butter, whisking constantly, until butter has melted. Lower heat to medium and continue whisking and reducing until a creamy, emulsified sauce forms (the time this takes will depend on your burner power and the size of your pan). If sauce over-reduces and breaks at any point, add one tablespoon cold water and whisk to emulsify it again. To keep sauce stabilized, you can add in a dash of cream, then whisk in lemon juice and parsley, season with salt and pepper, and remove from heat.
6. Arrange fillets on a platter and drizzle warm butter sauce over them. Serve immediately.