Tofu Puffs with Egg Gravy

Name: Tofu Puffs with Egg Gravy
Ingredients: 8 dried shitake mushrooms; 1 tablespoon Chinese light soy sauce (not low-sodium); 1 tablespoon oyster sauce; 1 1/2 teaspoons chicken bouillon powder; Kosher salt; 1/2 teaspoon sugar; 2 pinches ground white pepper; 3 tablespoons cornstarch; 1 tablespoon vegetable oil; 3 cloves garlic, minced; 1 teaspoon minced ginger; One 5-ounce package tofu puffs; 3 cups 1 1/2-inch-wide slices Napa cabbage; 1/2 cup snow peas, strings removed; 1 large egg, lightly beaten; 1/2 teaspoon toasted sesame oil; Cooked rice, for serving