Bring a large pot of salted water to a boil.
Meanwhile, whisk together the egg and 2 tablespoons cold water in a small bowl. Combine the chicken with 1/4 cup of the Parmesan in a second bowl.
Lay out a wonton wrapper on a work surface and top with 1 tablespoon of the chicken filling. Lightly brush the wrapper edge with the egg wash. Fold the wrapper in half to enclose the filling and pinch the edges to seal. Repeat with the remaining wrappers and filling.
Cook the dumplings in the boiling water until they float to the surface, 2 to 3 minutes. Serve with warm marinara sauce for dipping, or in bowls of warm chicken broth as a soup. Garnish with the remaining 1/4 cup Parmesan.
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