Preheat oven to 400°F (200°C). Line a large rimmed sheet pan with parchment paper. Arrange halved Roma tomatoes cut-side up alongside quartered onions. Drizzle with 2 tablespoons olive oil, sprinkle with sea salt, pepper, and oregano. Drizzle exposed garlic head with olive oil and wrap in foil on the pan. Roast for 35 to 40 minutes until tomato skins blister and edges caramelize.

While tomatoes roast, toss bread cubes with melted butter, grated garlic, and flaky sea salt on a separate baking sheet. Bake alongside tomatoes at 400°F for 10 to 12 minutes until crunchy and golden brown. Set aside.

Transfer roasted tomatoes, caramelized onions, and pan juices into a large Dutch oven or pot over medium heat. Squeeze the soft roasted garlic cloves out of the papery skins directly into the pot. Pour in the broth and balsamic vinegar. Bring to a gentle simmer for 10 minutes to marry flavors.

Remove from heat and stir in the fresh basil leaves. Use an immersion blender to puree the soup until velvety smooth. Alternatively, transfer in batches to a heat-safe high-speed blender.

Return soup to low heat. Stir in heavy cream and grated Parmigiano-Reggiano until melted and incorporated. Season with additional salt and cracked black pepper to taste.

Ladle hot soup into warm ceramic bowls. Swirl a spoonful of extra heavy cream on top, scatter fresh basil leaves, and crown with generous heaps of crunchy garlic croutons and shaved Parmesan.