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Step 1
Place flour, eggs and bread crumbs in three separate shallow bowls, and season each with salt and pepper. Crumble the oregano with your fingertips and add to the bread crumbs; mix well.
Step 2
Season fish with salt. Working with one piece at a time, dredge in flour (shake off excess), dip in egg (shake off excess) then dredge in bread crumbs, pressing to adhere. Transfer to a large plate.
Step 3
In a 12-inch nonstick skillet, melt 1 tablespoon of the butter in 1 tablespoon of the oil over medium heat. Add 2 fillets and cook until golden underneath, about 3 minutes. Flip fish and cook until golden on the second side and cooked through, 2 minutes longer. Transfer to a large paper towel-lined plate, and season with salt. Wipe out skillet. Repeat with 1 tablespoon of the butter, 1 tablespoon of the oil and the remaining fish.
Step 4
Wipe out skillet and melt the remaining 4 tablespoons butter over medium-low heat. Add shallot and cook, stirring occasionally, until shallot is softened and butter is golden brown, about 2 minutes. Stir in the garlic until fragrant, 30 seconds. Add the remaining 3 tablespoons oil, plus the lemon juice, parsley, capers and caper brine, and mix well. Transfer the brown-butter vinaigrette to a small heatproof bowl, and season with salt and pepper.
Step 5
Divide the fish, arugula and avocado among plates, and drizzle with the brown-butter vinaigrette. Serve with lemon wedges.