- Break leaves off 1 small head radicchio and tear into bite-size pieces; add to medium bowl. Trim bottom end of 1 head Belgian endive. Halve endive crosswise and separate leaves; transfer to bowl with radicchio. Add 2 cups baby arugula and toss to combine. Set aside.
- Tear off ¼ cup fronds from 1 small fennel bulb; set aside. Remove stalks, lightly trim base, and cut bulb in half lengthwise. Slice bulb into ½-inch-thick wedges; set aside.
- Pat 4 (6- to 8-ounce) skinless halibut fillets dry with paper towels and sprinkle with ½ teaspoon table salt and ¼ teaspoon pepper.
- Heat 3 tablespoons extra-virgin olive oil in 12-inch nonstick skillet over medium-high heat until just smoking. Add halibut and cook until browned on both sides, fish flakes apart when gently prodded with paring knife, and registers 130 degrees, 8 to 10 minutes, flipping halfway through. Divide halibut among 4 individual serving plates; set aside.
- Heat 2 tablespoons extra-virgin olive oil in now-empty skillet over medium heat until shimmering. Add fennel wedges and cook until golden, about 10 minutes, flipping once halfway through.
- Stir 1 minced large shallot and 3 minced garlic cloves into skillet with fennel and cook until softened and fragrant, about 2 minutes. Stir in ¾ cup dry white wine, ⅔ cup lemon juice, and 3 tablespoons water, and bring to simmer. Cook over medium heat until slightly reduced, about 2 minutes.
- Stir in 2 tablespoons Dijon mustard until well combined, then remove from heat. Season with salt and pepper to taste.
- Toss radicchio mixture, remaining 2 tablespoons oil, and remaining 1½ teaspoons lemon juice together in bowl. Divide fennel wedges evenly among reserved serving plates, then top fish in plate with radicchio mixture, creating mound on top of each fish fillet.
- Spoon sauce directly from skillet over top of each mound of greens, sprinkle with fennel fronds and ½ teaspoon flake sea salt, and drizzle with extra oil. Serve immediately.
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