Preheat a grill to medium. Brush the onions with vegetable oil on both sides and season with salt and pepper. Grill, covered and turning occasionally, until tender and charred, 15 to 20 minutes; roughly chop.
Combine the bourbon, ketchup, brown sugar and Worcestershire sauce in a large skillet. Simmer over medium-high heat until the sauce has thickened and the alcohol has burned off, about 3 minutes. Stir in the balsamic vinegar and add the onions. Season with salt and pepper. Serve warm or at room temperature.