Cut the stalks from the fennel bulbs; halve both the stalks and bulbs lengthwise. Using a mandoline, thinly slice one fennel bulb half. Transfer the slices to a small bowl of ice water and refrigerate for at least 30 minutes or up to 2 hours.
Meanwhile, cut the remaining 3 fennel bulb halves into 1-inch wedges. In a large skillet, heat 1 tablespoon of the olive oil. Add the fennel wedges in a single layer and add the chicken stock. Cover and cook over low heat, turning the fennel wedges a few times, until tender, about 25 minutes. Uncover, stir in the butter and season with salt and pepper. Keep warm.
Preheat the oven to 350°. In a large skillet, sear the fennel stalks over high heat, turning as necessary to char thoroughly on both sides, about 15 minutes.
Rub the fish fillets with oil, season with salt and pepper and arrange in the skillet on the charred fennel stalks. Bake on the top shelf of the oven for about 8 minutes, until the fish just flakes easily.
Drain the fennel slices and pat dry with a paper towel. Transfer them to a small bowl. Add the lemon juice and the remaining 1 tablespoon of olive oil, season with salt and pepper and toss.
Transfer the fish fillets to plates; discard the fennel stalks. Top the fish with the fennel salad, arrange the braised fennel wedges alongside and serve.