For the trout: Sprinkle the fish liberally with salt and pepper. Dredge one side of each fillet in the flour, and then in the eggs. Arrange the potato slices on the dredged side of each fillet in shingled, overlapping rows (they should look like scales).
When all of the fillets have been covered, place a large skillet over medium-high heat with 2 tablespoons butter or oil. Working in batches as needed, sear the fish, potato-side down, until the potatoes are deep golden brown and the fish is nearly cooked through, about 5 minutes. Gently turn the fish and finish cooking on the other side, about 1 minute. Keep the cooked fish warm on a platter covered with foil until all of the fillets are cooked.
While the fish is searing, place a small saucepan over medium heat with the wine and garlic. Bring up to a bubble and cook until the wine is reduced by half, 4 to 5 minutes. Remove the pan from the heat and add the remaining butter, swirling the pan to melt it. Season the sauce with the lemon juice and salt and pepper. Reserve warm.
For the brussels sprouts: Place a large skillet over medium-high heat with the butter or olive oil. Add the sprouts and cook, tossing occasionally, until they begin to wilt, 3 to 4 minutes. Add the pecans and raisins and continue cooking until the sprouts are tender, 3 to 4 minutes more. Season with salt and pepper.
To serve, divide the Brussels sprouts among serving plates. Slice the fillets on a diagonal and arrange over the Brussels sprouts. Garnish with the wine-butter sauce.
NotesThis recipe was provided by a chef, restaurant or culinary professional and makes a large quantity. The Food Network Kitchens chefs have not tested this recipe, in the proportions indicated, and therefore, we cannot make any representation as to the results.