Heat vegetable oil in a large stock pot over medium high heat. While the oil is heating, season the eye of round steak with salt and pepper. Once the oil is hot, add the beef in batches and brown for about 1-2 minutes. Remove the beef from the pot and set aside.
Next, add onions, celery and green bell pepper to the pot. Stir and allow the vegetables to simmer until they are tender, about 8-10 minutes.
Add the green onions and minced garlic. Continue to sauté for another minute.
Add the flour and stir everything around to get the ingredients coated. Cook for one more minute to get rid of the raw flour taste.
Add diced tomatoes with their juice and chicken or beef broth.
Add the beef back to the pot along with the leaves, mushrooms and red wine. If you don’t want to use wine, you can substitute with more chicken broth. Bring the mixture to a boil, then reduce the heat to low, cover the pot and allow everything to simmer for one hour, stirring occasionally. Season with salt and pepper to taste.
For the grits, combine the liquid ingredients in a large saucepan. Season with salt and pepper. Bring mixture to a low boil over medium high heat.
Stir in the grits. Reduce heat to medium low. Cover and simmer, stirring often, until liquid is absorbed and grits are tender, about 30-45 minutes.
Remove pan from heat. Stir in the butter. Season with additional salt and pepper if desired.