Preheat your oven to 425°F (220°C) with a rack in the center.
On a large, heavy-duty rimmed baking sheet, combine the sliced fennel, rinsed chickpeas, red onion wedges, artichoke hearts, cherry tomatoes, and sliced garlic. Drizzle with half of the extra virgin olive oil (2 tablespoons), the fine sea salt, and black pepper. Toss everything together until evenly coated, then spread into a single, even layer.
Thoroughly pat both branzino dry with paper towels, inside and out. Using a sharp knife, make 3 shallow diagonal scores on each side of the fish. Rub the remaining 2 tablespoons of olive oil all over the fish, ensuring it gets into the scores and the cavity. Place a few sprigs of parsley and one sprig of oregano inside the cavity of each fish.
Lay the conditioned branzino directly on top of the bed of vegetables on the baking sheet. Place the two lemon halves, cut-side down, on an open corner of the pan. Transfer the entire sheet to the preheated oven.
Roast for 18-20 minutes, or until the vegetables are tender and the fish is opaque and starting to flake.
Turn the oven to the high broil setting. Move the oven rack so the fish skin is about 6 inches from the heating element. Broil for 2-4 minutes, watching carefully, until the skin is beautifully blistered and crisp.
Carefully remove the baking sheet from the oven. Gently transfer the whole roasted fish to a large serving platter. With tongs, pick up the hot, charred lemon halves and squeeze their smoky, caramelized juice all over the vegetables remaining on the pan.
Toss the vegetables in the pan juices to create an instant, warm vinaigrette. Spoon this flavorful vegetable and chickpea mixture artfully around the fish on the platter. Garnish generously with fresh, chopped parsley and serve immediately.