Brown the short ribs: Pat the short ribs dry, season with salt and pepper, then sear them in olive oil over medium-high heat until deeply browned on all sides. Transfer to a plate.
Cook the vegetables: Add onion, carrots, and celery to the same Dutch oven. Cook for 6–8 minutes, then add garlic and cook for another minute.
Develop the base: Stir in tomato paste and cook for 1–2 minutes until slightly darkened.
Deglaze the pot: Pour in the red wine, scraping the browned bits from the bottom. Simmer for 8–10 minutes to reduce.
Braise the beef: Add beef stock, thyme, rosemary, and bay leaves. Return the ribs to the pot, cover, and cook at 325°F for about 2½–3 hours, or until fork-tender.
Shred the beef: Remove the ribs, discard the bones and excess fat, then pull the meat into bite-sized pieces.
Reduce the braising liquid: Strain about ½ cup of the cooking liquid and simmer it briefly until slightly concentrated.
Make the cheese crema: Warm heavy cream and butter over low heat. Gradually whisk in Grana Padano until smooth. Season with black pepper.
Prepare the gremolata: Mix parsley, horseradish, lemon zest, lemon juice, and a pinch of salt in a small bowl.
Toast the breadcrumbs: Cook panko in olive oil or butter until golden and crisp. Stir in parsley and Italian seasoning.
Cook the lumache: Boil the pasta in well-salted water until al dente. Reserve at least 1 cup of pasta water before draining.
Bring everything together: Add cooked lumache, shredded short rib, Grana Padano crema, and a little reduced braising liquid to a skillet. Toss gently over low heat.
Adjust the sauce: Add reserved pasta water a little at a time until the sauce becomes glossy and coats the pasta.
Finish the dish: Turn off the heat and fold in a small handful of arugula.
Serve: Spoon the pasta into bowls and finish with horseradish gremolata, herb breadcrumbs, extra Grana Padano, and freshly cracked black pepper.
