Place mushrooms in bowl; cover with hot water. Let stand 30 minutes.

Place carrot strips in small bowl. Add 1 teaspoon sugar and toss until mixed. Let stand 15 minutes, tossing occasionally.

Meanwhile, prepare Hoisin Peanut Dipping Sauce; set aside.

Drain mushrooms, reserving 1/2 cup liquid. Squeeze out excess water. Cut stems off mushrooms; discard. Cut caps into thin slices; set aside.

Heat wok over medium-high heat 1 minute or until hot. Drizzle 2 tablespoons vegetable oil into wok and heat 30 seconds. Add onion and stir-fry 1 minute. Stir in mushrooms, garlic and reserved mushroom liquid. Reduce heat to medium. Cover and cook 3 minutes or until mushrooms are tender. Uncover; add soy sauce, sesame oil and remaining 1 teaspoon sugar. Cook and stir mushroom mixture 3 to 5 minutes more or until all liquid has evaporated. Transfer mushroom mixture to medium bowl; set aside to cool slightly.

Add carrot strips and bean sprouts to mushroom mixture; toss lightly. Place 1 wrapper on work surface with corner at bottom, keeping remaining wrappers covered with plastic wrap. Drain mushroom mixture; place 3 tablespoons mixture on bottom third of wrapper. Brush edges of wrapper with some beaten egg.

To form spring rolls, fold bottom corner of wrapper up over filling. Fold in and overlap the opposite right and left corners to form 3-1/2-inch-wide log. Roll up filling to remaining corner and place spring roll on tray covered with plastic wrap. Repeat with remaining wrappers and filling.

Heat remaining 3 cups vegetable oil in wok over high heat until oil registers 375°F on deep-fry thermometer. Fry 4 rolls 2 to 3 minutes or until golden brown, turning once. Repeat with remaining rolls, reheating oil between batches. Drain on paper towels. Arrange on serving plate with bowl of Hoisin Peanut Dipping Sauce. Garnish as desired.

Combine peanut butter, water and soy sauce in small bowl until smooth. Stir in remaining ingredients. Pour into serving bowl.