Preheat oven to 425°F. Place tomatoes, onion, carrot, garlic, oregano, 1½ teaspoons of the salt, and ½ teaspoon of the pepper in a large bowl. Add 2 tablespoons of the oil; toss to coat. Spread vegetables, turning tomatoes cut side down, in a single layer on a large parchment paper-lined rimmed baking sheet. Roast in preheated oven until tender, 40 to 45 minutes.
Using tongs, peel off tomato skins; discard skins. Transfer tomatoes and remaining vegetable mixture, including all drippings, to a food processor; pulse until chunky, 5 to 7 times. Pour sauce into a medium saucepan; stir in sugar, and keep warm over low heat.
Heat a grill pan over high. Toss squid with remaining 1 tablespoon olive oil, 1 teaspoon salt, and ½ teaspoon pepper. Place squid on hot grill pan, and cook until charred on both sides, 1 to 2 minutes per side. Remove from pan, and slice crosswise into ¼-inch rings, including tentacles.
Toss cooked pasta, tomato sauce, and squid until well combined. Divide among 4 bowls, and top with Crispy Chickpeas. Serve immediately.