Add potatoes to a large pot or Dutch oven and cover with cool water by about 1-inch. Add 2 teaspoons salt, cover, and turn heat to HIGH. Watch closely and when water reaches a boil, reduce heat to MEDIUM and cook at a low boil for 15 to 20 minutes, or until potatoes are tender when pierced with a sharp knife, but not falling apart.
Meanwhile, place broccoli florets in a microwave-safe dish and add enough water to come up the sides about halfway. Microwave for 5 to 6 minutes or until fork tender (the time can vary depending on how big your florets are). Remove from the microwave, drain the water, turn out onto a cutting board and chop down into small ½-inch pieces.
Drain the potatoes and add them back to the warm, empty pot. Add the sour cream and about half of the milk,the onion powder, the remaining 1 teaspoon salt, and butter. Use a potato masher to smash the potatoes to a slightly chunky consistency (they should not be completely smooth), adding additional milk, as needed, to make them nice and creamy. Stir in the broccoli and cheese.
Taste and season with additional salt and pepper, as desired.