Melt the butter in a large Dutch oven or soup pot over medium-low heat.

Add the onion, garlic, red pepper flakes, ¼ teaspoon of the salt, and about 10 twists of black pepper.

Stir to combine, then cover the pot and cook until the onions have softened and are just starting to turn golden, about 8 to 10 minutes.

Prepare the broccoli by slicing off and discarding the rough bottom ends of the stalks and peeling the tough outer skin.

Slice the stalks into 1 to 2 inch pieces and reserve the florets for later.

Add the chunks of broccoli stalk and potato to the pot and stir.

Pour in 6 cups of water and the remaining teaspoon of salt. Raise the heat to high and bring to a boil, then reduce heat to medium-low and cover.

Simmer until tender, about 20-25 minutes. Meanwhile, chop the reserved florets.

Once tender, add half of the florets, cover, and cook for 3 to 5 minutes.

Blend the mixture until smooth and return to the pot.

Return to medium heat, add remaining florets, and cook for 4 to 7 minutes until bright green.

Stir in cheddar cheese until smooth and adjust seasoning.