Cut the beef into 1-by-2-by-1/4-inch strips. Put in a bowl and coat with the brown sugar, lime zest, 3 tablespoons of the soy sauce and some black pepper. Set aside for a few minutes while you make the sauce.
Whisk together the teriyaki sauce, cornstarch, sesame oil, rice vinegar, lime juice and remaining 3 tablespoons soy sauce in a bowl.
Heat a large skillet over medium-high heat. Add 1 tablespoon of the vegetable oil. Add the beef and stir-fry until brown around the edges, 2 to 3 minutes. Remove to a plate and keep warm.
Add the remaining 2 tablespoons vegetable oil to the skillet. Add the garlic, ginger and jalapeno and cook for 30 seconds. Add the green beans, carrots and bell pepper and stir-fry for 1 minute. Add the snow peas. Add the beef with its juices and the sauce and cook for another minute.
Serve over rice and garnish with scallions and sesame seeds.