In a large pot or container with a sous vide, heat 8 liters of water to 129°F/53°C.
In a medium bowl, combine the soy sauce, apple juice, mirin, sesame oil, sugar, and pepper.
Pack the flank steak into a sous vide bag with the marinade and the crushed garlic. You can marinate the steak in the fridge overnight or start cooking right away.
Cook the steak in the water bath for 10 hours.
Remove the cooked flank steak from its bag and pour all of the liquid into a medium pot. Add the sliced scallion whites (reserving the green part for garnish), carrots and onion. Bring to a boil and cook over medium heat for 10 minutes.
Meanwhile, in a small bowl, whisk the cornstarch into two tablespoons of cold water. Whisk the cornstarch slurry into the simmering sauce. Continue cooking for two more minutes, until the sauce is thickened. Strain the sauce and discard the solids.
Pat the flank steak dry with paper towels. In a large skillet over high heat, heat the butter and olive oil until it begins to smoke. Sear the flank steak for one and a half minutes per side, until a seared crust forms. Immediately remove the steak from the skillet and allow it to rest for two minutes.
Slice the steak thinly, top with the reserved scallion greens and serve with the sauce.