Prepare the grill for direct cooking over high heat (450° to 550°F).
Allow the brisket to stand at room temperature for 15 to 30 minutes before grilling. Generously brush or spray the brisket with olive oil. Sear over direct high heat, with the lid closed as much as possible, until well browned, about 6 minutes, turning once. Transfer to a braising pan, such as a 13-by-9-inch heavy-gauge aluminum pan.
Decrease the temperature of the grill to low heat (250° to 350°F). In a medium saucepan over medium heat, melt the butter with the olive oil. Add the onion, garlic, and caraway seed and cook for 5 to 6 minutes, stirring occasionally. Add the ketchup and broth, whisk, and bring to a simmer. Add the beer and balsamic vinegar. Return to a simmer, then pour the liquid over the brisket so it comes one-half to two-thirds of the way up the sides; don’t overfill the pan. Cover with foil and grill over indirect low heat, with the lid closed, until fork tender, 3 to 4 hours, turning twice.
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