Dust the alligator cutlets very lightly with flour and a little salt and white pepper. Heat the olive oil in a large skillet to just shy of smoking temperature and sauté the cutlets about a minute on each side. Remove from the skillet and keep warm.
In the remaining olive oil, sauté the onion, bell pepper, celery, fennel, and parsley until tender. Chop the tomatoes and add them to the skillet with the tomato juice, parsley, cayenne, lemon juice, and white wine. Bring to a simmer.
Return the alligator cutlets to the pan and cook them in the sauce for two or three minutes. Adjust the seasonings with salt and pepper. Serve with lots of sauce and hot buttered grits on the side.