Halve the potatoes and add to a large pot. Cover with water and bring to a boil over high heat. Reduce the heat to medium-low and cook the potatoes for 10 minutes while you trim and chop broccoli into 1/2-inch dice. 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, and transfer to a serving bowl, to serve.