In a saucepan of boiling water, cook the white bok choy portions for 1 minute; add the green portions and cook until wilted. Drain, rinse with cold water and squeeze dry; finely chop and transfer to a medium bowl. Stir in the pork, 1 tablespoon soy sauce, the oyster sauce, ginger, honey and 1 teaspoon sesame oil.
Arrange 10 wonton wrappers on a work surface and top each with a rounded teaspoon of the pork mixture. Moisten the edges and fold over to form half-moons; press the edges to seal. Working with 1 at a time, bring the tips together and pinch to form 10 bundles; repeat with the remaining wrappers and filling.
In a medium saucepan, bring the chicken broth, 1 cup water and the remaining 3 tablespoons soy sauce to a boil. Cover; remove from heat.
In a large pot of boiling water, cook the dumplings, stirring occasionally, until tender, about 5 minutes. Transfer to bowls. Add the snow peas to the broth, cook for 1 minute, then ladle over the dumplings. Drizzle with a few drops of sesame oil.