Cut the potatoes into equal pieces and add to a large pot. Add a couple extra inches of water to the pot, so when you add the broccoli everything will be covered. Add a teaspoon of salt; cover and bring to a boil over high heat. Reduce the heat to medium-low and cook the potatoes for 10 minutes.
While the potatoes are cooking trim and chop broccoli into ½-inch pieces (stalk and all).
Add the broccoli to the potatoes and cook for 5 minutes more. Drain and return to the hot pot to dry.
Stir in the milk, butter and cheese and mash until the cheese melts into potatoes and broccoli. Season with salt and pepper, to taste.