To a food processor combine edamame with tahini, garlic, ginger, white miso paste, soy sauce, and sesame oil. Blend until pureed. Add in matchstick carrots and pulse a few times until roughly chopped and combined.
Begin making your dumplings. Get a small bowl of cold water and set aside. On a clean workspace, lay out your dumpling wrapper. Add a small amount of edamame mixture to the center of the wrapper (~2 tsp).
Wet your fingertips and wet one half end up the dumpling wrapper. (It shouldn't be sopping wet but slightly damp).
Fold the bottom over the mixture and seal edges with fingertips.
Then pinch the wonton wrapper together using your the tip of your index finger and thumb along the seal of the dumpling.
Repeat this process with all dumplings until you use all filling (Mine made ~24 dumplings but it will depend on how much filling you place in each).
Warm ~1 tablespoon of sesame oil for a few minutes in a large non-stick frying pan over MEDIUM heat. Spread the oil evenly over the pan with a spatula.
Once oil is warm add enough dumplings in a single layer but don't overcrowd. You will likely need to cook ~3 rounds depending on the size of your pan (remember to add more oil each time you add more dumplings)
Cook on one side for 2 minutes, then gently flip and cook for another 2 minutes until the dumplings are slightly browned. Add ¼ cup of water, turn heat DOWN slightly, and cover pan to steam for 3 minutes or until most of the water is evaporated.
Optional: Make your simple dumpling sauce by whisking all ingredients together. Serve your finished dumplings with the dumpling sauce and sprinkle with freshly diced green onion.