Heat 1 tablespoon oil in large Dutch oven over medium heat until shimmering. Add andouille and cook, stirring occasionally, until lightly browned, 5–6 minutes. Using a slotted spoon, transfer andouille to bowl and set aside; reserve drippings in pot.
Add remaining tablespoon oil to pot with drippings. Add onion, garlic, 1 ½ teaspoons kosher salt, paprika and pepper flakes. Cook, stirring frequently, until onion has softened, about 3 minutes.
Add bouillon and cook, stirring constantly, 1 minute.
Add potatoes and stir to coat. Add 8 cups water; increase heat to high and bring to a boil. Reduce heat to medium-low and simmer, uncovered, until potatoes are just tender, about 10 minutes.
Transfer ¾ cup solids and ¾ cup broth to blender jar (try to skim as much grease off the top as possible as—use that as part of the broth measurement); set aside to cool slightly.
Stir collard greens into soup; simmer until greens have wilted and softened, about 10 minutes.
Add vinegar to blender with potato and broth mixture. Blend potato mixture until smooth, about 1 minute. Stir puréed mixture and browned andouille into soup. Season with salt and pepper to taste.
Serve with freshly grated Parmesan, optional.