01 - If using fresh peppers, roast them over an open flame or under a broiler until the skins are blackened and blistered on all sides. Transfer to a bowl, cover tightly, and let steam for 10 minutes. Peel away the charred skins, remove seeds and stems, then chop the peppers.
02 - Heat olive oil in a large pot over medium heat. Add the diced onion and cook, stirring occasionally, until soft and translucent, about 5 minutes. Add the minced garlic and cook for 1 minute until fragrant, being careful not to brown it.
03 - Add the chopped roasted red peppers and smoked paprika to the pot. Stir well to combine and cook for 2 minutes to allow the flavors to meld.
04 - Pour in the vegetable broth and bring to a gentle simmer. Cook for 15 minutes, stirring occasionally, to allow the peppers to fully soften and release their sweetness.
05 - Remove the pot from heat. Use an immersion blender to purée the soup until completely smooth and velvety. Alternatively, transfer the soup in batches to a countertop blender, blending until smooth before returning to the pot.
06 - Return the puréed soup to low heat. Stir in the whole milk and heavy cream, mixing thoroughly. Heat gently until just steaming—do not allow the soup to boil as this may cause the dairy to separate.
07 - Gradually add the shredded smoked Gouda cheese, a handful at a time, stirring constantly until fully melted and incorporated. Continue stirring until the soup is silky smooth. Season with additional salt and pepper to taste.
08 - Ladle the hot soup into serving bowls. Garnish with chopped fresh parsley and a sprinkle of additional shredded Gouda cheese, if desired. Serve immediately while hot.