Pulse ginger and garlic in food processor until finely chopped. Transfer half of mixture to bowl. Add asparagus, edamame, water chestnuts, 1-1/2 teaspoon soy sauce, and 2 teaspoon sesame oil to food processor; pulse until coarsely chopped.

Brush edge of each won ton wrapper with water. Place 2 teaspoons asparagus mixture in center of wrapper, then fold over filling to make a triangle-shaped dumpling. Repeat with remaining wrappers and filling.

Line 10-inch steamer basket with bok choy leaves. Place dumplings over bok choy, and cover. Set basket over large skillet filled with 2 inches boiling water, and steam 8 to 10 minutes, or until dumplings are translucent. Transfer dumplings and bok choy to serving plate, and keep warm. Repeat with remaining bok choy leaves and dumplings. For final round of dumplings, place broccoli florets in second steamer basket, and set over boiling water. Place dumpling basket on top, and steam 8 to 10 minutes. Remove dumpling basket, cover broccoli, and steam 2 to 4 minutes more.

Meanwhile, whisk together 1/4 cup soy sauce, rice vinegar, chile-garlic paste, and 2 tablespoons water.

Use skillet to heat remaining 2 teaspoon sesame oil over low heat. Add green onions and remaining ginger-garlic mixture, and saute 1 minute. Stir in broccoli, and cook 1 minute more. Remove from heat, and stir in sesame seeds, remaining 1-1/2 tablespoons soy sauce, and 2 teaspoons water. Serve dumplings with broccoli and sauce.