Prepare the filling: In a medium bowl, combine the shredded carrot, red bell pepper, mushrooms, sliced green onions, peanuts, cilantro, soy sauce, and cayenne pepper.
Mix until everything is well combined.
Assemble the wontons: Place about 1 rounded teaspoon of the vegetable filling onto the center of each wonton wrapper. To fold, bring the bottom corner up over the filling, tucking the point underneath.
Roll it up halfway, leaving the top corner exposed. Moisten one of the side corners with a dab of water.
Bring the two side corners together over the filling, overlapping them slightly and pinching the moistened corner to the dry one to seal securely.
Set the finished wonton aside on a lightly floured surface and repeat with the remaining wrappers and filling.
Cook the wontons: Bring the 2 quarts of water to a rolling boil in a large pot or Dutch oven. Carefully add the wontons one by one.
Once the water returns to a boil, reduce the heat and simmer gently, uncovered, for 5 minutes, or until the wontons are tender and float to the surface. Gently drain the wontons and rinse with cool water to stop the cooking process.
Simmer the broth: While the wontons are cooking, combine the vegetable broth, slivered green onions, and the piece of fresh ginger in a large saucepan.
Bring to a boil, then reduce the heat and simmer for about 2 minutes, just until the green onions are crisp-tender.
Remove and discard the piece of ginger.
Serve the soup: Carefully add the cooked wontons to the simmering broth.
Stir in the toasted sesame oil and heat through for about a minute. Ladle the soup and wontons into bowls and serve immediately.
Don't overfill the wontons.
Using about 1 teaspoon (5 ml / 0.2 fl oz) of filling per wrapper ensures they seal properly and don't burst while cooking.