Heat oven to 375°F. Lightly grease a 1.5–2 quart baking dish.
Beat cream cheese, sour cream, and mayonnaise in a bowl until smooth and fluffy.
Stir in Worcestershire, soy sauce, lemon juice, sugar, garlic, salt, pepper, and sriracha if using.
Fold in crab, green onions, half the mozzarella, and all the Parmesan until just combined.
Spread mixture into the dish and top with the remaining mozzarella.
Bake until hot and bubbling at the edges, 18–22 minutes.
Broil 1–2 minutes to lightly brown the top, watching closely.
Make wonton chips: Brush wrappers with oil, cut into triangles, and bake on a sheet at 375°F for 6–8 minutes until crisp.
Garnish the dip with extra green onions and serve hot with the wonton chips.