Heat 2 teaspoons oil in a large cast-iron skillet over medium-high heat. Sprinkle steak with salt and 1 teaspoon black pepper. Add one-third of steak to pan; cook 45 seconds on each side. Remove from pan; keep warm. Repeat procedure with remaining steak.
Add remaining 2 teaspoons oil to pan. Add white onion, celery, bell pepper, and garlic; sauté 3 minutes or until vegetables are tender. Sprinkle flour over vegetables, and stir well to combine. Add the diced tomatoes, tomato sauce, and olives; stir well to combine. Add broth; stir until well blended. Add remaining 1 teaspoon black pepper, basil, thyme, and hot pepper sauce.
Return steak to pan; bring to a boil. Reduce heat, and simmer 1 hour or until steak is tender. Stir in green onions and parsley. Serve over grits.
This is home food not "special occasion" but it's worth four stars. The gravy is really tasty and good, though I did use fire-roasted tomatoes with basil and garlic and a tomato sauce with basil and garlic in addition to the garlic cloves and basil I put in--so lots of nice flavor. Used thin cut eye-of-round and it was just right. Added more Tabasco and black pepper at table. I made only 8 oz. meat (for two people) but 50% of everything else, so we had some of the luscious sauce left over for lunch. Served with steamed cabbage wedges and grits.
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