Heat a gas grill to between medium and medium high. In a small bowl, combine 2 Tbs. of the brown sugar with the chile powder, dry mustard, and 1 Tbs. salt. Spread the chicken thighs on a large baking sheet and rub the spice mix all over them.
In a small saucepan, whisk the remaining 4 Tbs. brown sugar with the ketchup, soy sauce, cider vinegar, and Dijon mustard; bring to a boil. Lower the heat to a simmer and cook, stirring frequently, for 2 minutes to dissolve the sugar and blend the flavors. Remove from the heat and let cool slightly. Taste and add a pinch of salt, if needed. Stir in the Chipotle Tabasco, adding more to taste.
Spread the chicken thighs out on the grill, cover, and cook on the first side until they turn a deep reddish brown and begin to blacken in places (they'll also shrink and plump up), 4 to 5 minutes (rotate the thighs once 90 degrees on each side for the most even cooking). Flip the thighs and continue to cook on the second side until they're firm, deeply colored, and slightly blackened in places, 3 to 4 minutes. Using a brush, dab a generous amount of sauce over the top of the chicken, cook for 1 minute, flip, slather the other side with sauce, cook for 1 minute, and remove from the heat. Arrange on a platter and serve with any remaining sauce and the slaw if you like.
Toss 1/2 head green cabbage, cored and very thinly sliced (about 4 cups), with 3 Tbs. fresh lime juice, 1 tsp. granulated sugar, and 3/4 tsp. kosher salt. Let it sit for 20 to 30 minutes, tossing occasionally.