Cube the peeled potatoes into 1" pieces. Put the potatoes in a saucepan and cover with water. Bring to a boil and cook until fork tender. Right before the potatoes seem to be getting done add in the broccoli florets and cook for 3-4 minutes until the broccoli is bright green.

Drain the water off the potatoes/broccoli and add in the butter and cheese. Put a lid on the pan and give it a few good shakes to break up the potatoes a bit. Season with salt and pepper and serve.