Whisk soy sauce, vegetable oil, brown sugar, ginger, and garlic powder together in a large bowl. Pierce flank steak several times with a large fork. Place in the bowl and cover with plastic wrap. Let marinate in the refrigerator, at least 4 hours and up to overnight.
Bring a large pot of salted water to a boil. Cook udon noodles in boiling water, stirring occasionally, until tender yet firm to the bite, 13 to 14 minutes. Drain.
Heat a large skillet over high heat. Remove steak from marinade and cook until well-browned, about 2 minutes per side. Reserve marinade.
Preheat grill for medium heat and lightly oil the grate. Grill steak, basting with half of the reserved marinade, until internal temperature reaches 140 degrees F (60 degrees C) for medium or 150 degrees F (65 degrees C) for medium-well, at least 10 minutes per side. Slice steak thinly against the grain.
Combine remaining marinade, snow peas, broccoli florets, and mirin in the skillet. Cook and stir over medium-high heat until snow peas are tender, about 2 minutes. Add drained noodles; mix well to combine.
Divide noodle mixture among large bowls. Top with steak slices.