Heat 2 teaspoons sesame oil in a large skillet over medium heat. Add garlic and ginger and cook for about 30 seconds, or until fragrant, stirring constantly. Add chopped shiitake, water chestnuts, and carrot. Cook for four minutes, stirring frequently.

Add bok choy, green onion and miso paste and cook for 90 seconds, stirring frequently. Remove skillet from heat and scrape content into a large heat-proof bowl. Stir in pinch of sugar, rice wine vinegar, soy sauce, and remaining 2 teaspoons sesame oil. Taste mixture and season to taste, adding more soy sauce, sesame oil, sugar, or rice wine vinegar if desired.

Set up your gyoza-filling area: open wonton wrappers and then cover with a damp tea towel to prevent drying out. Fill a small bowl with water and set near the wonton wrappers. Working with about 2-3 at a time, lay wonton wrappers flat on your work surface, spoon a small amount of wonton filling into the center--about 2 teaspoons or so, maybe a little less. Then, dip your index fingers in the bowl of water and wet the edges of the wonton wrapper. Meet edges of wrapper around filling together and pleat one side, then press edges together so filling is contained. Make sure you don't let any air into the gyoza! If you need a bit of guidance here, check out this great tutorial for filling and sealing gyoza. Repeat until all filling has been used.

When all gyoza are formed, heat a large skillet with a tight-fitting lid over medium heat. Add about 2 teaspoons coconut oil or other neutral heat-friendly oil and heat thoroughly. Carefully add as many gyoza as can fit the bottom surface of the skillet, pleated side up. Cook for about 3 minutes or until golden brown on the bottom. Next, carefully pour in 1/4 cup water, using the lid to shield yourself from the splatter. When the steaming and splattering has died down, drizzle in 1/2 teaspoon sesame oil around the edge of the skillet. Cover with the lid and then quickly lower heat to keep the liquid at a bare simmer.

Check gyoza after a couple minutes - if the wrappers look nearly translucent, then remove the lid and raise heat slightly. Continue cooking until all the water has evaporated and only the oil is left - about 2 minutes. Gyoza should slide about freely if they're browned enough, but if not, simply remove the skillet from the heat, cover, and let sit for a moment, then remove gyoza with a spatula. Cook remaining gyoza in the same method.

To make sauce, combine soy sauce, rice wine vinegar, and green onion. Add Sriracha and/or sesame oil to taste. Serve gyoza immediately, with sauce.