How to Make It
Heat a large Dutch oven over medium-high heat. Add oil; swirl to coat. Add yellow onion, minced ginger, minced garlic, and jalapeño; sauté 4 minutes. Place onion mixture in a medium bowl.
Remove stems from mushrooms. Thinly slice mushroom caps; set aside. Add stems, stock, 2 cups water, vinegar, 1 teaspoon salt, black pepper, smashed garlic, and sliced ginger to pan; bring to a boil. Reduce heat; cover and simmer 45 minutes. Strain stock mixture over a bowl; discard solids. Return stock to pan over low heat, or follow freezing instructions.
Add remaining 1/4 teaspoon salt, chicken, and egg to onion mixture, stirring well. Working with 1 wonton wrapper at a time (cover the remaining wrappers to keep from drying), spoon about 2 teaspoons chicken mixture into center of wrapper. Moisten edges with water; gather up and pleat edges of wrapper around filling to form a purse. Repeat procedure with remaining chicken mixture and wrappers. Continue with step 4, or follow freezing instructions.
Add carrots to stock in pan; simmer 4 minutes. Add mushroom caps, dumplings, and peas; cook 8 minutes. Divide soup evenly among 8 bowls; sprinkle evenly with green onions.
HOW TO Freeze: Add vegetables to stock mixture; cool completely. Freeze stock mixture and dumplings separately in large ziplock plastic freezer bags. Thaw: Microwave stock mixture at MEDIUM (50% power) 3 to 4 minutes or until bag is pliable. Reheat: Pour stock mixture into a large Dutch oven; simmer 5 minutes or until thoroughly heated. Add frozen wontons; cook 3 minutes.