Whisk water, soy sauce, brown sugar, cornstarch, honey, and ginger together in a bowl until well combined.

Heat 1 tablespoon sesame oil in a large wok or skillet over medium-high heat. Add steak and cook and stir until browned, about 4 minutes. Move steak to the side of the wok. Add remaining tablespoon sesame oil to the center of the wok, followed by broccoli, snow peas, bell pepper, and garlic. Cook, stirring the vegetables, for 2 minutes.

Add reserved sauce to the wok and stir everything together. Bring to a boil and cook until the sauce thickens and the vegetables are crisp-tender, about 2 minutes.

Garnish with chopped green onions.