Combine boiling water and shiitakes in a bowl; cover and let stand 30 minutes. Drain the shiitakes in a colander over a bowl, reserving liquid. Slice shiitakes, discarding stems.
Heat 2 teaspoons oil in a large Dutch oven over medium-high heat. Add onion, shallots, ginger, and garlic; sauté 10 minutes or until lightly browned. Add the shiitakes, button mushrooms, sugar, and beef. Sauté 10 minutes, scraping pan to loosen browned bits. Add reserved mushroom liquid, 4 cups water, and the next 4 ingredients (water through consommé); bring to a boil. Cover, reduce heat, and simmer 50 minutes or until barley is tender. Stir in 1 teaspoon sesame oil.
Ladle 1 1/2 cups soup into each of 6 ovenproof soup bowls; top each serving with 2 bread slices and 2 tablespoons cheese. Broil 3 inches from heat 1 minute or until cheese melts. Serve immediately.
Note: This soup will freeze well for up to two months. Pour into an airtight container, leaving enough room for expansion (usually an inch or two at the top). To reheat, thaw completely in the refrigerator; then place contents in a saucepan over low heat, adding some liquid if necessary.