Bring a medium saucepan of water to a boil. Add edamame; cook 1 minute. Add ramen and cook 3 minutes. Drain, rinse with cold water and toss with sauce.
Heat a large nonstick skillet over medium-high heat. Crumble in ground beef and cook, breaking apart with a spoon, 4 minutes. Add sweet pepper, scallions and garlic. Cook 4 minutes.
Meanwhile, whisk together teriyaki sauce, lime juice, vinegar, sugar and red pepper flakes. Add to skillet and cook 2 minutes. Let everyone assemble their own wraps: Layer cucumber, meat mixture and mint into lettuce leaves. Roll up and eat with noodles on the side.