In a large, shallow bowl, mix soy sauce, apple juice, vinegar, sugar, garlic, and 1/4 teaspoon pepper. Add meat; toss to coat. Let marinate 15 minutes. Transfer meat to a plate; reserve marinade.
Heat 1 tablespoon oil in a 12-inch nonstick skillet over high heat. In two batches, cook meat until lightly browned, turning once, about 2 minutes per batch. Remove meat. Add 1/2 cup water to pan; stir up browned bits with a wooden spoon and pour into marinade; whisk in cornstarch (or arrowroot flour).
In same skillet, fry broccoli with remaining teaspoon oil over high heat until bright green and crisp, tossing often, 2 minutes. Add 1 cup water; cook until broccoli is tender, 6 to 8 minutes.
Stir marinade, add to pan, and bring to a boil. Cook, stirring, until thickened, 30 seconds. Return meat to pan; toss to coat. Season with salt and pepper. Serve immediately.