1. In a large skillet, heat oil over medium-high. Season short ribs with salt and pepper. In batches, cook until browned on all sides, about 10 minutes total. Transfer ribs to a 5- to 6-quart slow cooker. Pour off all but 1 teaspoon fat from skillet, and add onion and carrot. Season with salt and pepper and cook, stirring, until vegetables are soft, 3 minutes. Add 1/2 cup juice from tomatoes, stirring and scraping up any browned bits from skillet with a wooden spoon.

2. Transfer vegetables and liquid to slow cooker and add tomatoes, breaking them up as you go. Add herb sprigs; season to taste with salt and pepper. Cover and cook on high until meat is fork-tender, 6 hours.

3. With a slotted spoon, remove herb sprigs and transfer meat to a cutting board. With a ladle or large spoon, skim fat from cooking liquid (you'll remove about 1 1/4 cups). With two forks, shred meat and return to cooking liquid. Serve immediately, or let cool in liquid and reheat over medium before serving.