Heat the oven to 450°F, then quarter the tomatoes, or halve them if small, and place them cut-side up on a baking sheet with 2 tablespoons of olive oil, salt, and black pepper. Roast for 30 to 40 minutes, until the tomatoes are very soft with blistered, caramelized edges, then set the baking sheet on a wire rack.

Warm the remaining 1 tablespoon olive oil in a large stockpot over medium-high heat and cook the diced onion for 5 minutes, stirring now and then. The onion should soften and look translucent around the edges without taking on dark color.

Add the minced garlic, smoked paprika, and crushed red pepper flakes, then stir frequently for 2 minutes. The spices should sizzle gently and smell warm and fragrant, while the garlic remains pale rather than browned.

Pour in the vegetable broth, fresh basil, and all roasted tomatoes along with the juices collected on the baking sheet. Stir until the pot holds an even red mixture with the basil beginning to wilt into the hot liquid.

Blend the soup with an immersion blender until it reaches the texture you enjoy. As the olive oil emulsifies, the color shifts from bright red toward a velvety orange-red and the soup becomes glossy.

Taste and adjust with more salt, black pepper, or smoked paprika, then ladle the soup into bowls and add Parmesan if desired. The finished soup should taste rounded and smoky, with basil still noticeable beneath the roasted tomato flavor.