Make the beurre blanc: In a 10" skillet over medium-high heat, combine the wine, lemon juice, peppercorns, and garlic and cook until reduced to 2 tbsp., about 6 minutes. Set a fine strainer over a 1-qt. nonreactive saucepan set over low heat; strain wine mixture. Working with 14 tbsp. butter, whisk in 2 tbsp. at a time, waiting until the butter is emulsified before adding more, until sauce is slightly thick and smooth. Season sauce with salt and set aside in a double boiler over low heat to keep warm.
Make the garnish: Heat 2 tbsp. butter in a 10" nonstick skillet over medium-high heat. Add the onions and cook, stirring occasionally, until caramelized, about 12 minutes. Transfer onions to a plate; set aside. Add remaining butter to the skillet over high heat. Add capers and cook, stirring frequently, until crisp, 4 minutes. With a slotted spoon, transfer capers to the plate with the onions; set aside.
Make the fish: Julienne potatoes lengthwise using the narrowest setting of a French mandoline. (For similar results, use the large holes on a box grater.) Transfer potatoes to a strainer and rinse under cold water, turning and squeezing potatoes, until the water runs clear. Let drain for 5 minutes and pat dry with paper towels.
Pour oil into a 4-qt. saucepan to a depth of 2". Heat over medium-high heat until a thermometer reads 250°. Working in batches, cook the potatoes until just tender, about 3 minutes. Using a slotted spoon, transfer potatoes to paper towels. Divide into 8 portions and set aside.
Place the flour in a shallow dish. Whisk together the milk and egg in another shallow dish. Season the filets with salt and pepper and toss to coat in the flour; shake off excess flour. Dredge filets in the egg wash. Place each filet on top of a portion of potatoes and cover with another portion of potatoes, pressing lightly to adhere.
Heat oven to 200°. Heat 2 tbsp. oil in a 12" nonstick skillet over medium-high heat. Using a metal spatula, transfer 2 filets to the skillet. Cook, flipping once, until the potatoes are crisp and golden and fish is cooked through, about 12 minutes; transfer to a wire rack set over a baking sheet and keep warm in the oven. Wipe out the skillet and repeat with the remaining oil and filets.
To serve, spoon some of the beurre blanc onto a warm plate. Add a potato-crusted filet and sprinkle with toasted capers and caramelized onions.