Heat grill to medium (350° to 450°). Grill dogs until slightly charred all over, about 4 minutes, turning occasionally. Add buns and grill, turning frequently, until warmed and lightly charred, about 2 minutes.
Meanwhile, toss arugula and onion with oil, lemon juice, salt, and pepper in a medium bowl to lightly coat.
Smear mustard and cheese on buns, tuck in hot dogs, and top with arugula salad.
*Kobe beef hot dogs are available from marxfoods.com. Find Red Hawk cheese at cowgirlcreamery.com or Whole Foods Market.