Preheat oven to 400°F (200°C). Line a large rimmed baking sheet with parchment paper.
Arrange halved Roma tomatoes, cherry tomatoes, quartered onions, and whole garlic cloves in a single layer. Drizzle with 3 tablespoons olive oil and season generously with kosher salt and black pepper.
Roast for 25 to 30 minutes until tomato edges are lightly charred and onions are tender and caramelized.
Transfer roasted vegetables and all accumulated sheet-pan juices into a large Dutch oven over medium heat. Pour in vegetable broth and bring to a gentle simmer for 5 minutes.
Add fresh basil leaves. Use an immersion hand blender to puree soup directly in the pot until silky smooth and velvety.
Stir in heavy whipping cream and balsamic vinegar. Simmer on low for 2 minutes to meld flavors. Adjust salt and pepper to taste.
Prepare grilled cheese: Butter sourdough slices, layer with sharp cheddar, and skillet-toast over medium heat until golden brown and melted. Slice into 1-inch croutons and serve floating over hot soup bowls.