Pat veal shanks dry with paper towels and season generously with salt and pepper on all sides. Tie each shank around the middle with kitchen twine to keep the meat attached to the bone during cooking.

Heat olive oil in a large Dutch oven over medium-high heat. Brown the veal shanks on all sides, about 3-4 minutes per side, working in batches if necessary. Remove shanks and set aside.

Reduce heat to medium. Add diced onions, carrots, and celery to the pot. Cook, stirring occasionally, until vegetables soften and begin to caramelize, about 8-10 minutes. Add minced garlic and cook for 1 minute more.

Pour in white wine, scraping up any browned bits from the bottom of the pot. Let wine reduce by half, about 5 minutes.

Add diced tomatoes, tomato paste, bay leaves, and thyme. Stir to combine. Return veal shanks to the pot, arranging them in a single layer. The liquid should come about halfway up the shanks; add a bit of water if needed.

Bring to a simmer, then cover and transfer to a 325°F oven. Braise for 2 hours, turning shanks once halfway through, until meat is fork-tender and nearly falling off the bone.

While the ossobuco cooks, prepare the gremolata: finely mince parsley and combine with lemon zest and minced garlic in a small bowl.

Remove shanks from pot and keep warm. If the sauce is too thin, simmer it on the stovetop until reduced to desired consistency. Taste and adjust seasoning.

Serve shanks with sauce spooned over top, garnished generously with gremolata. Accompany with your choice of side.