1 tin (3 to 4 oz.) high-quality tuna packed in olive oil
1 anchovy fillet (or more, if you like things punchy)
¼ cup mayonnaise
¼ cup olive oil
Juice from ½ lemon (about 2 to 3 Tbsp.)
1 Tbsp. capers in brine, drained
1 clove garlic, peeled
1 tsp. ice-cold water
2 lbs. fresh, firm, flavorful tomatoes
2 Tbsp. fresh basil leaves, roughly torn
Fresh-ground black pepper
Make the tonnato sauce. In a blender or food processor, combine all ingredients (including the tuna oil), except water, and blend until the mixture achieves a pale, silken consistency, slightly thicker than cake batter, pausing and scraping down the sides as needed. Add ice-cold water and blend for another 10 to 15 seconds until the sauce takes on a slightly glossy texture. If the mixture feels too thick or lumpy, add more ice-cold water, 1 tsp. at a time, until you reach the desired consistency.
Transfer tonnato to a serving bowl, cover tightly with plastic wrap, and transfer to the refrigerator to set for 30 minutes or up to 3 days.
Prepare the tomatoes. Wash tomatoes and remove their stems. Using a very sharp knife, slice tomatoes latitudinally into ½-inch rounds. On a serving platter, arrange a single layer of tomato slices, and dust sparingly with kosher salt. If more tomatoes remain, arrange them on top of the lower layer and sprinkle each slice with a few grains of salt.
Just before serving, remove the prepared tonnato sauce from the fridge and stir to incorporate any oil that may have separated. Spoon about half the tonnato sauce over the tomatoes, reserving the remainder for another day (or for slathering on bread, or eating furtively with a spoon). Garnish with basil leaves and several coarse grinds of black pepper.