Make the marinade: Heat the vegetable oil in a small saucepan over medium heat. Add the thyme, garlic and 1/2 teaspoon pepper and cook, stirring occasionally, until the garlic is softened, about 4 minutes. Remove from the heat, carefully add the bourbon and then return to the heat. Simmer until reduced by more than half, about 5 minutes. Pour into a large bowl; whisk in the brown sugar, mustard and 1/2 teaspoon salt. Let cool to room temperature.
Add the steak to the bowl, toss well and let marinate in the refrigerator 2 to 6 hours.
Soak twelve 8- to 10-inch skewers in water for 30 minutes to prevent scorching. Preheat a grill to medium high. Drain the steak, discarding the garlic and thyme. Thread the steak and red onion onto the skewers. Season all over with salt and pepper.
Oil the grill grates. Grill the kebabs, turning occasionally, until the onion and steak are well charred, 4 to 7 minutes for medium doneness.