In a small bowl, combine the paprika, brown sugar, dry mustard, sage, 1 tsp. salt, and 1 tsp. pepper. Set aside 2 tsp. to use in the braise and sprinkle the remaining rub all over the ribs. Cover and refrigerate for at least 2 hours but preferably overnight.
Prepare a gas grill for direct grilling over medium-high heat. Grill the ribs until nicely browned, 3 to 5 minutes per side. Transfer to a platter, let cool briefly, and then tie each rib with 3 or 4 loops of butcher’s twine.
Prepare the grill for indirect grilling. In an 8-quart heavy-duty pot, cook the bacon over medium heat, stirring occasionally, until it just starts to crisp, 2 to 3 minutes. Add the onions and cook, stirring occasionally, until soft and beginning to color, 6 to 8 minutes. Add the garlic, carrot, and the reserved spice rub and cook for about 1 minute more. Add the broth, beer, vinegar, bay leaves, and caraway seeds. Bring to a boil, reduce the heat, and simmer, uncovered, for 10 minutes.
Nestle the ribs into the braising liquid. Set the pot on the grill over the cool zone. Cover the pot, close the grill lid, and cook until fork-tender, about 1-1/2 hours, turning the ribs halfway through cooking.
Transfer the ribs to a tray. Strain the braising liquid into a heatproof vessel, such as a Pyrex measuring cup, and let sit until the fat rises to the top. Discard the solids. Skim off and discard the fat. Keep warm.
Prepare the grill for direct grilling over medium-low heat. In a small bowl, stir the mustard, brown sugar, and Worcestershire sauce. Stir in just enough of the braising liquid, 1 Tbs. at a time, to produce a glaze thin enough to easily brush on the ribs. Brush one side of the ribs with the glaze and grill glazed side down until bubbly and beginning to darken, 3 to 5 minutes. Brush the other side, flip the ribs, and grill until the glaze is bubbly and beginning to darken, an additional 3 to 5 minutes.
Remove the string from the ribs, put them on a warm platter, and drizzle with the remaining braising liquid.
Make Ahead Tips
You can make this dish through the braising step the day before serving. Just cool and wrap the meat and braising liquid separately. Refrigerate overnight and glaze the next day. To spread out the work even more, you can season the meat the day before you braise.
To use a charcoal grill: To sear the spareribs, build a medium-high fire: Ignite about 5 quarts of charcoal (80 to 100 briquettes), using either a chimney starter or an electric starter. When the charcoal is burning well, spread it out over the surface of the charcoal grate and put the cooking grate in place. Let the charcoal burn down until it's coated with gray ash. To test the temperature, hold your hand about two inches above the cooking grate; when you can hold your hand there for 2 to 3 seconds (a medium-high fire), you're ready to sear the spareribs.
After searing, divide the coals evenly, banking them against two sides of the grill (use long-handled tongs to move the coals). Put the cooking grate in place; if your cooking grate has hinged sections, position them over the charcoal. Place an oven thermometer on the grate over the cooler area, close the lid, and let the coals burn until the thermometer reads about 350ºF. When ready, place the pot over the cool zone and braise as directed above. Check the thermometer every 20 minutes, replenishing the charcoal as necessary to keep the temperature between 325ºF and 375ºF. If the coals are still burning well, you can simply add a handful of unlit coals on top. Otherwise, you'll need to add lit coals.
To glaze the spareribs, build a second fire just as you did in the searing step, but let the coals burn down until you can hold your hand two inches above the cooking grate for 5 to 7 seconds (a medium-low fire). Continue with the glazing, as directed in the recipe.
Note: Grill-braising on charcoal could blacken your stainless steel pot. If you don’t want to deal with cleanup, use a disposable aluminum pan and cover it tightly in aluminum foil, both of which can be recycled.
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Photo: Scott Phillips