1. Combine 1 cup boiling water and shiitake mushrooms in a bowl; cover and let stand 20 minutes. Drain mushrooms in a colander over a bowl, reserving liquid. Rinse mushrooms. Remove and discard stems; cut each cap into quarters. Set aside.
2. Combine soy sauce, wine, and sugar in a small bowl, stirring with a whisk.
3. Heat a small saucepan over medium-high heat. Add 1 teaspoon oil and ginger to pan; sauté 30 seconds. Add reserved mushrooms; sauté 1 minute. Add reserved mushroom liquid and broth; bring to a boil. Cover, reduce heat, and simmer 20 minutes.
4. Combine oyster sauce and cornstarch in a small bowl, stirring with a whisk; stir into mushroom mixture. Bring to a boil, stirring constantly. Cook 1 minute or until thickened. Remove from heat; keep warm.
5. Heat a wok or large skillet over high heat. Add remaining 1 tablespoon oil to pan. Add garlic; stir-fry 10 seconds. Add bok choy; stir-fry 2 minutes or until bok choy begins to soften. Add lettuce; stir-fry 2 minutes or until lettuce wilts. Stir in mushroom mixture and soy sauce mixture; cook 3 minutes or until bok choy is tender.