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.