Step 1: Build the broth. Heat the olive oil in a medium or large soup pot over medium heat. Add the chopped onion, sliced ginger and smashed garlic. Cook, stirring often, for about 5 to 6 minutes until the onions soften and turn lightly golden at the edges and the mixture smells fragrant. Pour in 1 cup of the broth and scrape the bottom of the pot to release any browned bits. Add the remaining broth, dried shiitake mushrooms if using, soy sauce, rice vinegar and sesame oil. Bring to a gentle boil, then reduce to a low simmer, cover loosely and cook for 30 to 60 minutes to develop flavor.
Step 2: Mix the wonton style filling. While the broth simmers, place the vegan ground, panko breadcrumbs, prepared vegan eggs, soy sauce, grated ginger, minced chives, rice vinegar, black pepper and sugar in a mixing bowl. Stir or use clean hands to mix until everything is evenly combined and the mixture feels slightly sticky. If it seems too wet, sprinkle in a little extra panko. Let the filling rest for 5 to 10 minutes so the breadcrumbs hydrate and the mixture firms up slightly.
Step 3: Strain the broth and cook the filling. Set a fine mesh sieve over a clean pot and carefully pour the hot broth through, pressing on the solids to extract as much liquid as you can. Discard the cooked aromatics and mushrooms. Return the clear broth to the stove and bring it back to a gentle simmer. With slightly damp or lightly oiled hands, roll the filling into small teaspoon sized balls and drop them into the simmering broth. Work in batches if needed so they have room to cook. Cover and simmer for about 12 to 15 minutes, until the filling is firm and cooked through.
Step 4: Add the greens. Stir the chopped baby bok choy into the simmering broth. Cook for 2 to 3 minutes, until the stems are just tender and the leaves turn bright green. The bok choy should still have a bit of bite.
Step 5: Add the wonton wrappers. Add the sliced or quartered wonton wrappers a few at a time, stirring gently as you go so they do not clump together. Simmer for 2 to 3 minutes, just until the wrappers are soft and silky but still hold their shape. Taste the broth and adjust the seasoning with extra soy sauce for more depth or a splash of rice vinegar for brightness.
Step 6: Serve. Ladle the hot broth, filling, bok choy and wrappers into warm bowls. Top each bowl with sliced green onion and a drizzle of chili oil. Serve your deconstructed vegan wonton soup right away for the best texture and flavor.