Combine 4 tablespoons butter, the capers, parsley, lemon zest and juice, 1/2 teaspoon salt and a few grinds of pepper in a bowl with a rubber spatula; set aside. Bring a medium saucepan of salted water to a boil.
Melt the remaining 1 tablespoon butter in a large nonstick skillet over medium-high heat. Add the bread pieces, season with salt and cook, stirring occasionally, until toasted, about 4 minutes; transfer to a plate.
Wipe out the skillet; add the vegetable oil and heat over medium-high heat. Season the fish on both sides with salt and pepper and add to the skillet, skin-side down. Cook until the skin is browned and crisp, about 4 minutes. Carefully flip and continue cooking until just cooked through, about 1 more minute; transfer to a plate.
Meanwhile, add the haricots verts to the boiling water and cook until crisp-tender, about 4 minutes. Drain and season with salt and pepper. Divide among plates and sprinkle with the croutons. Add the fish and top with the lemon-caper butter; serve with lemon wedges.