1. Heat the oven to 400°. Cut the Brie into thin slices and divide the cheese among four small ovenproof dishes or ramekins.
2. Put the potatoes in a large saucepan of salted water. Bring to a boil and simmer for 10 minutes. Add the broccoli florets to the pan and simmer until the potatoes and broccoli are tender, about 5 minutes longer. Remove the broccoli with a slotted spoon and drain on paper towels. Drain the potatoes and, if using new potatoes, cut them in quarters when cool enough to handle.
3. Meanwhile, toss the mushrooms with the oil and salt. Put the mushrooms on a baking sheet and roast until browned and tender, turning once, 10 to 15 minutes. Remove the pan from the oven and then turn the oven off.
4. Put the dishes of cheese in the oven and leave until the cheese just melts, 5 to 10 minutes. Meanwhile, pile the potatoes, broccoli, mushrooms, and cocktail onions on individual plates. Serve each portion of melted Brie immediately, along with the vegetables for dipping.
Wine Recommendation: The wines of the Loire Valley in France are frequently overlooked by serious imbibers, but its elegant cabernet-franc-based reds are often more suitable at the table than more robust cabernet sauvignons. Go for a fruity Chinon or Saumur-Champigny.
Notes: Cocktail onions, most frequently found in your martini glass, can stand on their own as a pickle. Their slight crunch and pleasant tartness make them surprisingly tasty with Brie.