This dish is named after the Italian fishing port Livorno, site of one of the annual spring tuna harvests, and the recipe's not carved in stone tablets anywhere: a quick Google session will reveal that there is a wide range of interpretations being presented as "Tuna Livornese."
As usual, I went to Elizabeth David's Italian Food, and found that her version is typically succinct: "Fry slices of ... fresh tunny in oil. Drain them. Stew them for 30 minutes in a fresh tomato sauce flavoured with garlic. Garnish with plenty of parsley."
Mulling over all the Googled versions and Elizabeth's, I decided that the basic ingredients are tomatoes, onions, garlic and the tuna. Basil, surprisingly to me, is apparently not one of the usual suspects for this dish. Other additions range from olives to capers to red pepper flakes, and the sequence and methods of cooking are all over the map.
Working with Elizabeth's outline, the first thing I realized was that I'm not that big on stewing fresh tuna for 30 minutes...so I made a fresh tomato sauce -- skimping on the garlic, which I feel can overpower the fresh tomato and delicate tuna flavor, and adding basil, which I feel enhances both -- and then I seared the tuna steaks and brought the fish and sauce together on the plate.
This was an easy, quick (for me) and very successful combination of several of my favorite fresh local ingredients, with satisfying results.
4 tuna steaks, 6 oz each, thick-cut
2 or 3 tomatoes, a little over a pound, peeled, cut in rough 1" pieces
1 medium yellow onion, sliced thin
1/4 C celery, chopped
1/4 C flatleaf parsley
1 clove garlic, minced
1 C basil, minced
1 bay leaf
salt, hot sauce to taste
Sauté the onion, celery and parsley slowly in 1 tablespoon olive oil, stirring occasionally, until the onion turns translucent, about 8 minutes. Add the tomatoes, garlic and bay leaf, give a stir, then cover and simmer 15 minutes. Season with a dash of hot sauce if desired.
Wash the steaks and pat dry. Salt the steaks all over. In a sauté pan, heat a tablespoon of olive oil over high heat to just below the smoke point and then sear the steaks about 2 minutes on a side.
Just before serving, stir 3/4 of the basil into the sauce. Remove bay leaf. Correct seasoning, being careful with salt since the fish was salted. Place a bed of sauce on the serving plate and then the tuna atop that. Garnish with a spoonful of the sauce and scatter on the rest of the basil.