Preheat the oven to 300 degrees. Heat a large Dutch Oven over medium-high heat.
Season short ribs with salt and pepper. Sear in 1 Tablespoon oil until browned on all four sides, 3-4 minutes per side. Transfer to a plate.
Add remaining Tablespoon oil to the pot. Sauté carrots and shallots/onions for 5-7 minutes, then add garlic for 1-2 minutes.
Stir in tomato paste for 1 minute. Add red wine, bring to a simmer, then reduce for 5 minutes.
Add beef broth, San Marzano tomatoes, bay leaves, and fresh thyme. Nestle short ribs into the pot, cover, and braise in the oven for 4 hours.
Remove short ribs, discard bones, and shred the meat. Refrigerate the meat and sauce separately.
Remove bay leaves and thyme stems from the sauce. Refrigerate the sauce for at least 4 hours, or overnight, until fat solidifies.
Scrape off fat, reheat sauce, and mash tomatoes/vegetables. Add shredded short ribs and a drizzle of lemon juice. Taste and adjust seasoning.