Place the steak in a medium bowl and add 3 tablespoons of the dressing to the steak. Reserve remaining dressing and set aside. Let the steak marinate, turning every few minutes for at least 20 minutes, up to 1 hour.
While the steak is marinating, bring a pot of water to a boil. Remove from heat, add noodles, and cover. Let sit, stirring every minute to keep from sticking, until noodles are al dente, 5–8 minutes. Drain and rinse under cold running water. Drain well and set aside.
Combine cabbage and cucumbers in a large bowl; season lightly with salt. Massage with your hands to release excess water. Drain off any water that pools in bottom of bowl.
Pat steak dry and season with salt. Heat remaining oil in a large cast iron skillet over medium-high. Add steak and cook, turning once, until browned all over, about 5-6 minutes per side for medium. Let rest 5 minutes before cutting into semi-thin slices against the grain.
Add cabbage mixture, noodles, steak, ½ cup mint, and ½ cup peanuts to bowl with reserved dressing. Toss to combine; taste and season with salt if needed. Top with more mint and peanuts before serving.