Instructions

Cut 4 large tomatoes in half horizontally. Using your fingers, scoop out seeds and pulp and transfer to fine-mesh strainer set over large bowl. Press on pulp to extract juice; discard pulp.
Combine 1½ teaspoons salt and 1 teaspoon sugar in bowl. Sprinkle cut side of tomato halves evenly with salt mixture. Place tomato halves cut side down in colander set over bowl of tomato juices and drain for 5 minutes. Measure out ⅓ cup tomato juices and stir in 1 tablespoon extra-virgin olive oil and 1 teaspoon sherry vinegar. Discard any remaining juices.
Meanwhile, pulse 6 ounces ½-inch rustic bread pieces in food processor until pieces resemble coarse crumbs that are around ⅛ inch (some larger pieces are OK).
Heat 1 tablespoon olive oil in 12-inch nonstick skillet over medium-high heat until shimmering. Add tomatoes, cut side down, and cook until edges of tomatoes have some spotty browning, about 3 minutes. Transfer tomatoes, cut side up, to 13 by 9-inch broiler-safe baking dish. 
Add 3 tablespoons olive oil and bread crumbs to now-empty skillet and cook over medium-high heat, stirring frequently, until browned and crisp, 5 to 7 minutes. Remove skillet from heat and push bread crumbs to sides to clear center; add remaining 1 tablespoon olive oil, 1 teaspoon herbes de Provence, and ½ teaspoon minced garlic to clearing. Using residual heat of pan, cook, stirring constantly, for 10 seconds. Transfer bread crumbs to bowl, add ¼ cup grated Parmesan, ¼ teaspoon pepper, and remaining ½ teaspoon salt and toss to combine. 
Adjust oven rack 7 inches from broiler element and heat broiler. Fill each tomato with ¼ cup bread crumb mixture, pressing into cavities. Drizzle about 1½ teaspoons tomato juice mixture over bread crumbs, then evenly pack tomatoes with remaining bread crumb mixture, mounding excess. Sprinkle with remaining ¼ cup grated Parmesan. Broil until topping browns, 1 to 2 minutes. Let cool for at least 5 minutes. Serve warm or at room temperature.