Slice the top round into 4 equal size pieces then pound with a meat mallet to approximately 1/2" thick.
Heat a large heavy pan to a touch higher than medium heat with 2 tablespoons of neutral oil. Pat the steaks very dry and liberally season with salt and pepper on both sides. Dredge the steaks into the flour and shake off the excess. Add the steaks to the hot pan and sear until well browned on both sides then remove the steaks to a plate.
Add the remaining oil to the pan along with the onions, celery, carrot, and a pinch of salt. Saute until very soft (about 10-15 minutes) then add the garlic and cook for another 2 minutes or until lightly golden and fragrant.
Add the beef stock and bring to a boil while scraping the pan with a wooden spoon to dislodge any brown bits. After 2 minutes, add the tomatoes, coarse black pepper, worcestershire sauce, and sugar to the pan and stir together. Bring the sauce to a lively simmer. Taste the sauce and add salt and pepper to taste.
Add the steaks to the pan and cover with the lid. Cook over a low simmer until very tender (90-120 minutes). Flip the steaks every 30 minutes to ensure even cooking. If the sauce starts to dry out during the braising process, just add a few ounces of water to the pan.
Once the steaks are very tender, turn off the heat and taste test one more time, adjusting salt and pepper if required. There will be a good amount of fat in the sauce that has risen to the top. If you like, you can spoon it out, or use paper towels. Simply lay the paper towels on top of the sauce and lift. The fat will adhere to the paper towels.
Top with minced parsley and serve with grated cheese and crusty bread to mop up the amazing sauce. Enjoy!