Preheat the oven to 350 degrees and make sure your racks are placed so you can add the large pot or dutch oven since you will cook this all in one pot.
Remove any extra fat on the eye of round roast.
In a large pot or dutch oven heat 2 Tbs oil. Salt and pepper each side then add to the pot. Sear each side to trap in the juices for 2-3 minutes. Remove the beef from the plate.
In the same pot add the onion and garlic. Sautee for 2-3 minutes. To the onions now add the wine, beef stock and tomato paste. Stir to combine all the liquids together and cook down the tomato sauce so there are no more clumps.
Add the beef back into the pan. Finally add the carrots.
Place a top over the pot and into the oven this goes for 3 hours. Check the roast at 2 to 2-1/2 hours to make sure that there is still liquid in the pot. Adding 1/4 cup at a time as needed.
Take the roast out of the sauce and shred or slice it as you like. Add the meat back into the sauce and combine. Serve with noodles or mashed potatoes. Enjoy!