Heat 1 tablespoon oil in Dutch oven over medium-high heat until shimmering. Add sausage and cook, stirring occasionally, until lightly browned, 4 to 5 minutes.
Transfer sausage to bowl and set aside. Reduce heat to medium and add onion, garlic, 1¼ 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. Add kale to pot and simmer for 10 minutes. Stir in chorizo (and beans if using) and continue to simmer until greens are tender, 8 to 10 minutes longer.
Add remaining 3 tablespoons of oil to soup in blender and process until very smooth, 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 with crusty bread.