For the fluke Milanese: Line a baking sheet with parchment. Whisk the eggs in a shallow dish. Mix together the panko and Parmesan in a separate shallow dish.
Sprinkle the fluke fillets on both sides with salt, and working in batches, dip each thoroughly in the eggs and then thoroughly in the breadcrumbs. Carefully place the coated fish fillets in a single layer on the prepared baking sheet and refrigerate until ready to cook.
For the salsa verde: Whisk together the capers, caper brine, olive oil, honey, hot sauce, parsley, tarragon, anchovies, anchovy oil, lemon zest, lemon supremes and juice in a medium bowl. Taste for seasoning, then set aside.
Finish the fish: Heat the canola oil in a large skillet over medium heat until it starts to smoke lightly. Add the fish in a single layer and cook on the first side until golden brown, 2 to 3 minutes. Using a metal fish spatula, gently flip the fillets and cook until firm to the touch, an additional 2 to 3 minutes. Remove to a paper towel-lined plate to drain slightly. Season again with salt.
Pour out all but 1 tablespoon of the oil from the fish skillet. Add the shallots and a pinch of salt and cook until they become translucent and tender, 3 to 5 minutes. Turn the heat to low and add the vermouth and 1 tablespoon butter and continue cooking, 1 to 2 minutes. Add the remaining 3 tablespoons butter and lemon juice, season with salt and stir until the butter is completely melted, about 1 minute, then remove from the heat.
For the spinach: Add 1/4 cup water and 1 tablespoon butter to a medium skillet over medium-high heat. Simmer until slightly reduced and glaze-like, about 3 minutes. Add the spinach with a pinch of salt and lightly wilt, 2 to 3 minutes.
Place a spoonful of the spinach on the center of each of two plates and spoon the shallot sauce around it. Place the fish on top of the spinach on an angle and top with a drizzle of shallot sauce and salsa verde. Finish with a sprinkle of flaky sea salt.
To cut into supremes: After zesting, peel and segment the lemon with a paring knife, separating the fruit from the membranes. Work over a bowl and squeeze the empty membranes to release the juices.