For the noodles: Bring a large pan of lightly salted water to a boil. Snap the noodles into 3-inch lengths, add to the pan, and cook for 3 to 4 minutes. Drain, then rinse under cold water and drain again. Set aside until assembling.
For the dressing: Combine all of the dressing ingredients in a food processor and pulse to blend and dissolve the sugar. Set aside until assembling.
For the beef: Preheat grill to medium high. Combine all of the marinade ingredients in a food processor or mixing bowl; then pour over skirt steak. Marinate for 2 hours. Turn occasionally. Wipe off excess marinade and then place skirt steak onto grill and cook to medium rare; 5 to 8 minutes. Transfer to a cutting board. Let meat rest for 5 minutes before slicing.
To assemble: When slicing the beef, hold your knife at a 45-degree angle and cut the steak crosswise into very thin slices. Transfer the meat with its juices into a large mixing bowl and toss with a spoonful of the dressing. Except for the cabbage, toss beef with the remaining cut salad vegetables, noodles, chopped mint and peanuts; then toss with more dressing.
Arrange cabbage leaves on a platter or onto individual plates, then spoon the steak mixture on or around the salad. Sprinkle salad with a generous pinch of toasted sesame seeds.