If you can't find packaged broccoli slaw, whirl broccoli stems in a food processor, using the grater attachment. Throw in a carrot and some red cabbage if you like.
Heat grill to high (450° to 550°). Mix fish sauce, sherry vinegar, 2 tbsp. soy sauce, the brown sugar, garlic, and lemongrass; add chicken and toss to coat.
Mix remaining 1 tbsp. soy sauce, the rice vinegar, oil, mirin, ginger, and chile. Toss with broccoli slaw and green onions.
Grill chicken, basting with marinade and turning once, until cooked, about 10 minutes. Serve chicken over slaw.
*Find in the Asian-food aisle.
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