This is a great summery barbecue beef sandwich, although I eat it all year long, and the pickle slaw is incredibly refreshing. When you eat this, you’ll need a lot of napkins.
Preheat the oven to 425 degrees. Season the short ribs generously with salt and pepper.
In a large Dutch oven or similar pot, heat the oil over medium-high heat. When hot, sear the short ribs on all sides until the meat is crusty and nicely browned, 5-6 minutes total. Lift the short ribs from the pot and set aside on a platter.
Add the onion and garlic to the pot and cook, stirring, until nicely browned, about 5 minutes. Stir in the ketchup, broth, vinegar, lemon juice, Worcestershire sauce, sugar, and paprika and bring to a simmer. Taste and season with salt and pepper.
Return the ribs to the pot, cover, and roast in the oven until the beef is fork-tender, about 2 hours. Remove the pot from the oven and set aside to allow the meat to cool.
When the ribs are cool enough to handle, lift them from the pot, shred the meat using your fingers or a fork, and put it in a bowl. You will have about 3 cups of shredded short ribs. Set aside until ready to use.
Strain the cooking juices through a sieve into another bowl. Refrigerate the juices until you can skim and discard the fat that rises to the top, 20-30 minutes.
Spoon as much of the defatted cooking juices into the rib meat as needed to moisten it. You will need at least ¼ cup and perhaps more.
Reheat the meat on high power in the microwave until heated through, 1-2 minutes.
Preheat the broiler and toast the rolls, cut sides up, for 1-2 minutes, until nicely browned.
Pile equal amounts of meat on the bottom half of each roll and top with pickle slaw. Put the top half of the roll on each sandwich, press gently, and serve.