Place the flank steak on a cutting board and use a meat mallet or the back of a skillet to pound the steak until it is 1/2-inch thick. Remove and discard the parchment paper, then slice the steak against the grain into 1/4-inch-thin pieces. Add the steak to a large bowl and sprinkle with the arrowroot starch, salt, and pepper. Toss to coat.
Heat the oil in a large nonstick skillet set over medium-high heat. Working in batches, add the steak in a single layer and cook, undisturbed, until it forms a deep brown crust, 3 to 4 minutes. Flip and cook for 3 to 4 minutes more. Transfer the cooked steak to a plate and set aside. Repeat until all the steak is cooked.
Add the garlic, ginger, sesame oil, and red pepper flakes to the skillet and cook, stirring, for 1 minute. Pour in the beef broth and bring to a simmer, about 5 minutes. Continue to simmer for about 2 minutes, using the back of a spoon to scrape up any browned bits from the pan.
Return the steak to the skillet. Stir in the green onions, coconut aminos, rice vinegar, and fish sauce and bring the sauce to a simmer. Cook, stirring often, until the sauce has thickened, 3 to 5 minutes. Remove from the heat and set aside.