Cook the leeks in the butter with ¾ teaspoon salt and ¼ teaspoon pepper in a large pot over medium-high heat, stirring occasionally, until softened, 4 to 6 minutes.
Add the vegetable broth, broccoli, and potato. Simmer until tender, 18 to 22 minutes.
Stir in the spinach until wilted. Transfer the mixture to a blender and puree until smooth. Top with the sour cream and black pepper to taste before serving.