Preheat a large soup pot over medium heat. Cook the onions, stirring occasionally, until softened, about 8 minutes. Add garlic and cook, stirring frequently, 1 to 2 minutes. Add mushrooms and cook, stirring occasionally, until softened, about 10 minutes.
Add tomato paste, smoked paprika, cayenne, nutritional yeast, and chorizo, and stir to combine. Cook 1 to 2 minutes to heat the chorizo and allow flavors to mingle. Add potatoes, broth, salt, and pepper, and bring to a boil.
Add chickpeas and reduce heat to a simmer. Cook for about 15 minutes or until potatoes are fork tender. Stir in kale, remove from heat, and let sit for a few minutes.
Serve with a generous dollop of vegan yogurt. We loved this soup with homemade cashew yogurt, but store-bought yogurt or sour cream is also fine, as long as it's 100% plain and unsweetened.
Store leftover soup in the refrigerator for up to 5 days. Can also be frozen for extended storage.Nutrition information is an estimate and will vary depending on exact ingredients/brands used. Based on 6 servings per batch of soup.