Lay the slices of beef flat on your work surface and season with salt on both sides.
Place 1 piece prosciutto over each slice of meat, followed by 3 or 4 sticks each of carrot and celery at one short end of the meat. Roll the meat around the vegetables, forming a medium-tight involtino. Use kitchen twine or a couple of toothpicks inserted flush with the meat to keep the roll closed.
Heat the olive oil in a small skillet over medium heat. When the oil begins to shimmer, add the involtini and brown them on all sides, about 5 minutes total. Remove the rolls from the pan and set aside.
Add the garlic to the same pan and cook, stirring occasionally, until it just turns golden, about 5 minutes. Stir in the tomatoes. Add the wine and cook until the alcohol aroma dissipates, about a minute.
When the sauce begins to simmer, return the involtini to the pan. The meat should be mostly covered by the tomato sauce.
Cook, covered, until the meat is fork-tender, about 1/2 hour, checking occasionally to be sure the meat is at least two-thirds submerged and adding water if necessary.
Season with salt and pepper.
Serve immediately or allow the dish to rest in the refrigerator for up to 3 days to allow the flavors to develop.