Roast the vegetables: Preheat the oven to 400°F (200°C). Cut the bell peppers in half lengthwise and remove the stems and seeds. Arrange them cut-side down on a large baking sheet along with the onion quarters and the whole garlic head (with the top sliced off). Drizzle everything with olive oil and season generously with salt and black pepper. Roast for 30-35 minutes, or until the pepper skins are blistered and charred, the onions are tender, and the garlic is soft.
Transfer the peppers to a bowl and cover tightly with plastic wrap, a plate, or place them in a sealed container for 10 minutes. Peel off and discard the skins, then roughly chop the peppers.
Prepare the soup base: Melt the butter in a large pot or Dutch oven over medium heat. Add the roasted peppers, onion, garlic (squeezed from their skins), and tomato paste. Cook for 1–2 minutes, stirring frequently, until the tomato paste darkens slightly. Stir in the smoked paprika, oregano, and cayenne (if using) and cook for 30 seconds, just until fragrant.
Simmer: Add the crushed tomatoes, broth, sugar, and season with salt and black pepper. Stir well and bring to a gentle boil. Reduce the heat to low and simmer for 15–20 minutes to allow the flavors to meld.
Blend: Using an immersion blender, blend the soup until completely smooth. Alternatively, carefully blend in batches using a countertop blender, then return the soup to the pot.
Add the cheese: Reduce the heat to low (do not let the soup boil). Gradually add the grated smoked Gouda, one handful at a time, stirring continuously until completely melted before adding more. Stir in the cream, if using. Taste and adjust the seasoning with additional salt and black pepper as needed.
Serve: Ladle into bowls and serve warm with crusty bread, homemade croutons, grilled cheese, or an extra sprinkle of smoked Gouda.
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Notes
Peeling the peppers: Peeling the peppers is optional, but recommended. Removing the skins creates a smoother, silkier soup with a cleaner roasted pepper flavor. If you don't mind a slightly more rustic texture, you can skip this step, the soup will still be delicious.
Use freshly grated Gouda: Pre-shredded cheese contains anti-caking agents that can prevent it from melting smoothly.
Add the cheese over low heat: Avoid boiling the soup after adding the cheese, as high heat can cause it to become grainy or separate.
Garlic: Roasting the whole head creates a sweet, mellow garlic flavor without overpowering the soup. Simply squeeze the softened cloves from their skins after roasting.
Cream: The soup is delicious without cream, but adding a splash of cream or half-and-half, or even coconut milk makes it even silkier and richer.
Sugar: Depending on the sweetness of your peppers and tomatoes, you may not need the sugar. Taste the soup before adding it and only use it if needed to balance the acidity.
Consistency: If the soup is too thick, stir in a little extra broth until it reaches your preferred consistency.
Storage: Refrigerate in an airtight container for up to 4 days or freeze for up to 3 months. Thaw overnight in the refrigerator and reheat gently over low heat, stirring occasionally.