Heat oil in a Dutch oven over medium-high heat. Add chicken and cook, stirring occasionally, until just cooked through, about 6 minutes. Transfer to a plate with tongs.
Add ginger and garlic to the pot and cook until fragrant, about 10 seconds. Add sherry and cook until mostly evaporated, scraping up any browned bits, 1 1/2 to 3 minutes. Add broth and water, increase heat to high and bring to a boil. Boil for 5 minutes. Add soy sauce, chile sauce and mustard greens (or chard) and cook until the greens are tender, about 3 minutes. Return the chicken and any accumulated juices to the pot and cook until heated through, 1 to 2 minutes.