In a large pot or Dutch oven, melt butter over medium heat. Add onion and cook until softened, about 5 minutes.
Add garlic and red pepper flakes and cook for 1 minute more.
Add potato and broccoli stalks and cook for 5 minutes.
Pour in water and bring to a boil. Reduce heat and simmer until potatoes and broccoli are tender, about 15-20 minutes.
Add broccoli florets and cook for 5 minutes more.
Remove from heat and carefully blend with an immersion blender until smooth. (Alternatively, transfer to a regular blender in batches.)
Stir in 6 ounces of grated cheddar cheese until melted.
Season with salt and pepper to taste.
While soup simmers, preheat oven to 350°F (175°C). Line a baking sheet with parchment paper.
Place remaining cheddar cheese on the baking sheet and bake for 5-7 minutes, or until melted and bubbly.
Let cool slightly, then break into crumbles.
Serve soup topped with cheddar cheese crumbles.