Heat olive oil or butter in a large soup pot over medium heat.
Add the diced onion and cook for 5 to 7 minutes, stirring occasionally, until softened and translucent.
Add minced garlic and cook for 30 seconds, stirring constantly until fragrant.
Stir in the tomato paste and cook for 1 minute.
Add drained roasted red peppers, vegetable broth, dried basil, smoked paprika, black pepper, and red pepper flakes if using.
Bring the soup to a gentle simmer. Reduce the heat to low and simmer for 15 minutes, stirring occasionally.
Remove the soup from the heat. Blend until smooth with an immersion blender. If using a countertop blender, blend carefully in small batches and allow steam to escape.
Return the soup to low heat. Stir in the heavy cream.
Add the freshly grated Gouda a handful at a time, stirring until melted after each addition. Do not boil the soup after adding the cheese.
Taste the soup and add salt if needed. Stir in lemon juice or balsamic vinegar if you want a slightly brighter flavor.
Serve hot with fresh basil, extra Gouda, croutons, or toasted bread.