Pat the veal shanks dry with paper towels. In a shallow dish, combine the flour, salt, and pepper. Dredge each veal shank in the flour mixture, shaking off any excess.
In a large skillet over medium-high heat, melt the butter. Add the veal shanks in a single layer (work in batches if needed) and brown for 4-5 minutes per side, until a deep golden crust forms. Transfer the browned shanks to the slow cooker.
Reduce the heat to medium. Pour the white wine into the skillet and use a wooden spoon to scrape up any browned bits from the bottom of the pan. Let it simmer for a minute.
Add the chopped onion, carrot, and celery to the skillet. Cook, stirring occasionally, for about 5 minutes, until the vegetables have softened.
Stir in the undrained diced tomatoes, beef broth, and fresh thyme sprigs. Bring the sauce to a gentle simmer.
Pour the entire contents of the skillet over the veal shanks in the slow cooker. Cover and cook on low for 7-8 hours or on high for 4-5 hours, until the meat is very tender.
Just before serving, make the gremolata. In a small bowl, combine the chopped parsley, lemon zest, and minced garlic. Mix well.
Carefully remove the veal shanks from the slow cooker and place them on serving plates. Spoon the sauce over the meat and sprinkle with the fresh gremolata before serving.