Bring a small saucepan of water to a boil. Add mushrooms; cook 4 minutes. Place noodles in a medium bowl; add boiling water (with mushrooms). Let stand 10 minutes or until noodles are tender; drain. Pick out mushrooms; discard stems. Finely chop mushroom caps. Snip noodles with scissors to chop.
Place pork in a mini food processor; pulse 5 or 6 times or until ground. Spoon into a medium bowl. Add mushrooms, noodles, carrot, fish sauce, and pepper; mix well to incorporate (your hands work best for this).
Working with 1 wrapper at a time (cover remaining wrappers to prevent drying), place wrapper on a work surface, starchy side up. Moisten edges with water. Spoon about 1 tablespoon filling into center of wrapper. Bring 2 opposite corners together; pinch points together to seal. Bring remaining 2 corners to center; pinch to seal. Pinch 4 edges together to seal. Place dumpling on a baking sheet lined with plastic wrap coated with cooking spray. Repeat with remaining wrappers and filling.
Heat a large nonstick skillet over medium-high heat. Add 2 teaspoons oil; swirl to coat. Arrange 9 dumplings in pan; cook 2 minutes or until well browned. Add 1/3 cup water to pan; cover and cook 5 minutes. Uncover and cook 1 minute or until liquid evaporates. Remove dumplings from pan, and keep warm. Repeat procedure with remaining 1 1/2 teaspoons oil, remaining 9 dumplings, and remaining 1/3 cup water.