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We prefer collard greens, but kale can be substituted. Serve this soup with hearty bread and, for added richness, a final drizzle of extra-virgin olive oil.
Heat 1 tablespoon oil in Dutch oven over medium-high heat until shimmering. Add chorizo and cook, stirring occasionally, until lightly browned, 4 to 5 minutes. Transfer chorizo to bowl and set aside. Reduce heat to medium and add onion, garlic, 1 ¼ teaspoons salt, and pepper flakes and season with pepper to taste. Cook, stirring frequently, until onion is translucent, 2 to 3 minutes. Add potatoes, broth, and water; increase heat to high and bring to boil. Reduce heat to medium-low and simmer, uncovered, until potatoes are just tender, 8 to 10 minutes.
Transfer ¾ cup solids and ¾ cup broth to blender jar. Add collard greens to pot and simmer for 10 minutes. Stir in chorizo and continue to simmer until greens are tender, 8 to 10 minutes longer.
Add remaining 3 tablespoons oil to soup in blender and process until very smooth and homogeneous, about 1 minute. Remove pot from heat and stir pureed soup mixture and vinegar into soup. Season with salt and pepper to taste, and serve. (Soup can be refrigerated for up to 2 days.)