Bring a small saucepan of water to a boil. Add mushrooms and let it cook for 4 minutes. Place noodles in a medium bowl and add the boiling water, mushrooms included. Let it stand until the noodles are tender, about 10 minutes, then drain. Pick out the mushrooms and discard the stems. Finely chop up caps. Snip the noodles with scissors to chop.

Place pork in a mini food processor and pulse until it's fully ground; 5 or 6 times should do it. Spoon the pork into a medium bowl. Add mushrooms, noodles, carrot, fish sauce, and pepper and mix well to incorporate (your hands are actually better for this than a spoon or whatever).

Place one wrapper at a time on a work surface, starchy side up (cover remaining wrappers to keep them from drying out). Use water to moisten the edges, then spoon about 1 tablespoon of filling into the center. Bring 2 opposite corners together and pinch the points together to seal. Bring remaining 2 corners to the center and seal them the same way. Pinch all 4 edges together to fully seal. Place the dumpling on a baking sheet lined with plastic wrap coated with cooking spray. Do this with the rest of the wrappers and filling.

Heat a large nonstick skillet on medium-high heat. Add 2 teaspoons of oil and swirl around to coat. Arrange 9 of the dumplings in the pan and cook until well browned, about 2 minutes. Add 1/3 cup water to pan, then put the cover on and let it cook for 5 minutes. Uncover and cook an additional minute to get the liquid to evaporate. Remove the dumplings from the pan and keep warm. Keep doing this with remaining 1 1/2 teaspoons oil, remaining 9 dumplings, and remaining 1/3 cup water.