Heat butter in large Dutch oven over medium-high heat. When foaming subsides, add broccoli, onion, garlic, dry mustard, cayenne, and 1 teaspoon salt. Cook, stirring frequently, until fragrant, about 6 minutes. Add 1 cup water and baking soda. Bring to simmer, cover, and cook until broccoli is very soft, about 20 minutes, stirring once during cooking.
Add broth and 2 cups water and increase heat to medium-high. When mixture begins to simmer, stir in spinach and cook until wilted, about 1 minute. Transfer half of soup to blender, add cheddar and Parmesan, and process until smooth, about 1 minute. Transfer soup to medium bowl and repeat with remaining soup. Return soup to Dutch oven, place over medium heat and bring to simmer. Adjust consistency of soup with up to 1 cup water. Season to taste with salt and pepper. Serve, passing extra Parmesan separately.