Thinly slice the meat across the grain and set it aside.
In a mixing bowl whisk together beef broth, soy sauce, balsamic vinegar, honey, and cornstarch. Set aside.
Heat 1.5 tablespoons vegetable oil in a wok or a large skillet set over medium-high heat.
Add ginger and cook for 15 seconds.
Add peppers and onions and continue to cook for 4 minutes.
Stir in garlic and bok choy; add mushrooms and cook for 3 more minutes, or until veggies are crisp tender.
Remove vegetables from the skillet and set aside.
Return skillet to heat and add remaining oil.
To the heated oil add the sliced beef; cook for 2 to 3 minutes, or until meat is browned.
Push beef to one side of the skillet.
Whisk previously prepared sauce just enough to mix it. Pour the sauce in the center of the skillet; cook for a minute, or until thickened and bubbly.
Return vegetables to the skillet and stir all ingredients together; coat with the sauce and continue to cook for 1 more minute.
Remove from heat.
Garnish with green onions.
Serve over rice or noodles.