To make the wontons:Combine all the wonton ingredients (except for the wrappers) in a large bowl. Use a potato masher to mash the mixture, breaking down the pieces of shrimp until you get small chunks. Continue mashing until you get a smooth texture but not to the point where it's a paste.
Keep the wontons in the package so they don't dry out or place a slightly moist towel over them. Place a wonton wrapper in the palm of your hand and spoon about 1/2 tablespoon of filling onto the center of each one. Have a bowl of water handy. Dip your other finger in the water and trace around all sides of the wrapper to moisten them. You make can fold the wontons several ways according to your personal preference: Make a simple triangle by folding one corner over the other and pinching to seal the edges; for a little fancier design pull each corner of the triangle down so they overlap and press to seal (resembling a nurse's cap); or place the wonton in the palm of your hand with the filling in it, wet the edges and draw all the edges together and twist to seal (resembling a drawstring money bag). Repeat the process for the remaining wontons. SEE NOTE about freezing them.
To make the soup: Heat the oil in a stock pot and fry the garlic and ginger for 1-2 minutes until soft, then add the chicken broth, tamari, rice vinegar, sesame oil, and salt. Bring to a boil, cover and simmer for 10 minutes. Add the carrots and cabbage and simmer for another 5 minutes. Add the wontons, yellow miso paste and green onions and simmer for another 5 minutes (6-8 minutes if using frozen wontons). Add salt to taste.Note: Alternatively some people cook the wontons separately in water, then place the cooked wontons in a bowl and ladle the broth over to serve the wonton soup. Do whichever you prefer.
Ladle the soup into bowls and serve hot.
Once the wontons are assembled (before they're cooked), they can be frozen. Spread them out single layer on a non-stick cookie sheet and put them in the freezer. Once frozen you can place the wontons in a ziplock bag or airtight container and freeze for up to a couple of months. Note: From a texture standpoint it is not recommended to freeze the wontons if you're using previously frozen shrimp.