I freely admit that the tonnato preparation is an idea that takes some getting used to: It means that a meat, usually veal, is topped with tuna sauce—and it’s usually served cold. But before you say “Ugh” and turn the page, please give it a try. It’s really a wonderful combination of flavors. Instead of veal, I prefer the lighter taste of turkey paired with the relatively strong sauce, and I like this dish warm, not cold.
Preheat the Oven to 375 degrees F. Place the turkey in a baking pan and rub the oil over the turkey. Sprinkle with the dried herbs and 1 teaspoon each of salt and pepper. Pour 2 cups of the broth around the turkey and bake until it is just cooked through, about 30 minutes. Let the turkey cool in the pan for 10 minutes, then transfer the turkey to a cutting board. Using a large, sharp knife, cut the turkey diagonally into ½-inch-thick slices. Return the turkey slices to the baking pan and coat with the pan juices.
In the bowl of a food processor, blend the tuna, lemon juice, capers, anchovy paste, and the remaining 1 tablespoon of broth until creamy, about 1 minute. Transfer the tuna mixture to a medium bowl, and stir in the mayonnaise. Season the tuna sauce with salt and pepper to taste.
Arrange the turkey slices on a platter and pour the tuna sauce over. Garnish with the parsley and serve.