Bring 1 cup water to boil in small saucepan. Add porcini mushrooms. Remove from heat and let stand until mushrooms soften, about 20 minutes. Using slotted spoon, transfer porcini to work surface. Coarsely chop porcini. Reserve porcini and soaking liquid.
Melt butter in heavy large saucepan over medium heat. Add shiitake mushrooms and sauté until tender, about 3 minutes. Add shallot; sauté 1 minute. Add porcini and soaking liquid, leaving any sediment from liquid behind. Increase heat to high. Simmer until liquid evaporates, about 3 minutes. (Can be made 8 hours ahead. Cover and chill.)
Toss Brie with cornstarch in large bowl to coat. Add wine to mushrooms. Bring to simmer over medium heat. Add cheese to mushrooms in 3 batches, whisking after each addition until cheese melts before adding more. Continue whisking until mixture is smooth and just begins to simmer (do not boil). Season to taste with salt and pepper. Transfer fondue to fondue pot. Set pot over candle or canned heat burner. Serve with chicken, vegetables and bread.