1. Melt the butter in a heavy kettle and saute the onion and garlic until tender but not browned. Add the potatoes and saute for 4 minutes longer.
2. Add the broth, thyme, salt, and pepper, bring to a boil, cover, and simmer for 15 minutes, or until the potatoes are tender.
3. Add the broccoli flowerets, cover, and cook for 7 minutes, or until the flowerets are barely tender.