Place onions in a microwave-safe, small bowl and cook in the microwave for 2 minutes to soften. Set aside.
In a medium pot, melt butter. Stir in milk, chicken or vegetable broth, and condensed cheddar cheese soup, Mix well. Bring up to a simmer and continue to simmer, uncovered for 10 minutes. Stir often.
Mix in softened onions, broccoli florets, carrots, celery, garlic powder and a little black pepper, to taste. Continue to simmer until all vegetables are tender, about 15-20 minutes. Stir often.
To serve: Add 1¼ cups soup to each bowl. Top each with ¼ cup shredded cheese. Mix into soup.