Instead of using the broiler to get those nice charred bits and cook the sauce, just turn your oven up to about 450 degrees F for the last five to 10 minutes of cooking.
To complete the experience, thicken the sauce first. While the chicken cooks, reduce the BBQ sauce by simmering it in a saucepan over medium-low heat for 20 to 30 minutes.
Only reduce it by 25% to 50%, as any more than that could concentrate it too far and make it too salty. Then, just brush on the thickened sauce for that last round of high heat.