Preparation

Braise the ribs

• Position an oven rack about 6 inches from the broiler element and heat the broiler on high.
• Arrange half of the ribs in a single layer in a broiler pan (without the top rack). Broil, turning occasionally, until browned on all sides, about 15 minutes. Transfer to a platter and discard the fat in the pan.
• Put the pan over high heat. Add 1 bottle of the beer and bring to a boil, scraping up the browned bits from the bottom of the pan with a wooden spatula. Transfer the mixture to a large roasting pan. Repeat with the remaining ribs and beer.
• Clean the broiler pan. Reposition the rack to the center of the oven and lower the oven temperature to 350°F.
• Heat the oil in a 12-inch skillet over medium heat. Add the onion, carrot, celery, and garlic and cook, stirring occasionally, until the onion softens, about 5 minutes. Transfer to the roasting pan holding the beer mixture; stir in the broth, soy sauce, and red pepper flakes. Spread the vegetables in an even layer and arrange the ribs on top (it’s OK if they’re crowded).
• Cover the roasting pan with heavy-duty aluminum foil, crimping the edges to create a tight seal. Braise in the oven for 30 minutes. Turn the ribs, re-cover the pan, and braise until the ribs are tender when pierced with a small knife, 60 to 80 minutes more.
• Line the clean broiler pan with aluminum foil and transfer the ribs to the broiler pan. Tent loosely with more foil while you make the sauce.

Make the sauce

• Strain the braising liquid through a fine sieve into a medium saucepan; discard the solids. Let stand for 5 minutes; skim off and discard the fat on the surface. Bring to a boil over high heat. Lower the heat to maintain a steady simmer and cook, stirring often, until the liquid has reduced to 1-1/2 cups, about 20 minutes. Whisk in the brown sugar, mustard, and vinegar and raise the heat to return to a boil. Then lower the heat to maintain a simmer and cook, stirring occasionally, until reduced to 1-1/2 cups, 10 to 15 minutes.

Finish the ribs

• Leaving the rack in the center of the oven, heat the broiler on high.
• Brush the tops of the ribs with about 1/2 cup of the sauce; broil until glazed and sizzling, 5 to 7 minutes. Flip the ribs, brush with another 1/2 cup of sauce, and broil until the other side is glazed and sizzling, 5 to 7 minutes more. Transfer to a serving platter, drizzle with the remaining sauce, and serve.

Make Ahead Tips

You can braise the ribs and make the sauce up to 1-1/2 hours ahead and keep them at cool room temperature. Brush the ribs with sauce and finish them under the broiler just before serving.