Preheat oven to 425°F. Cut peppers in half, remove stems and seeds, and place cut-side down on a parchment-lined baking sheet. Drizzle with 1 tablespoon olive oil and roast 25-30 minutes until blistered.
Transfer peppers to a bowl, cover with plastic wrap, and let steam for 10 minutes. Peel off skins, chop the flesh, and set aside.
Heat remaining olive oil in a Dutch oven over medium heat. Cook onion, carrots, and celery for 5-7 minutes until softened. Add garlic and cook until fragrant, about 1 minute.
Stir in tomato paste and cook for 1 minute. Add diced tomatoes, roasted peppers, broth, thyme, red pepper flakes (if using), salt, and pepper. Bring to a boil.
Reduce heat to low, partially cover, and simmer for 15-20 minutes. Remove fresh thyme sprigs if used.
Remove from heat and blend with an immersion blender until completely smooth. If using a countertop blender, work in small batches with the lid vented.
Return soup to low heat and stir in heavy cream, Parmigiano Reggiano, and fresh basil until cheese is melted. Adjust seasoning to taste.
Ladle into bowls and garnish with extra cream, cheese, and fresh basil leaves. Serve warm.