Whisk the brown sugar, soy sauce, cornstarch, 2 tablespoons water and 2 teaspoons sesame oil in a bowl. Add the steak and toss.
Heat a large skillet over high heat; add 2 teaspoons sesame oil. Add the scallions, cucumber and cabbage; stir-fry until the vegetables start softening, about 2 minutes. Add the carrots and 1/4 cup water; cook, stirring, until the water evaporates and the vegetables are tender, 2 to 3 minutes. Transfer to a bowl; wipe out the skillet.
Heat the remaining 2 teaspoons sesame oil in the skillet over medium-high heat. Add the ginger and jalapenos; stir-fry 30 seconds. Transfer the steak to the skillet using a slotted spoon; stir-fry until browned, 2 to 3 minutes.
Add the steak marinade back to the skillet and cook, stirring, until thick, about 2 minutes. Return the vegetables to the skillet and heat through. Serve over rice.