Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, about 20 minutes. Drain and cool slightly. Thinly slice potatoes.
Place a steamer insert into a saucepan and fill with water to just below the bottom of the steamer. Bring water to a boil. Add broccoli, cover, and steam until tender, 2 to 3 minutes.
Heat vegetable oil in a skillet over medium heat; cook sliced potatoes until lightly browned, about 5 minutes. Stir broccoli, cream of chicken soup, milk, salt, and garlic powder into potatoes; cook until heated through, about 5 minutes. Add Cheddar cheese to broccoli-potato mixture; cook until cheese melts, about 1 minute.
