Heat butter in large Dutch oven over medium-high. Add broccoli, onion, garlic powder, dry mustard, cayenne, and 1 tsp salt. Cook, stirring frequently, until fragrant and soft, about 6 mins.
Add 1 cup water and baking soda. Simmer, cover, and cook until broccoli is very soft, about 20 mins, stirring once.
Stir in chicken broth and 2 cups water. Increase heat to medium-high. When simmering, add spinach and cook until wilted, about 1 min.
Blend half the soup with cheddar and Parmesan until smooth, about 1 min. Pour blended soup into a bowl. Repeat with remaining soup.
Return blended soup to Dutch oven over medium heat. Simmer gently. Adjust consistency with up to 1 cup additional water if needed.
Season with salt and black pepper. Serve hot with extra Parmesan on the side.