To make the sauce, combine the tuna, anchovies, capers, white wine vinegar, lemon juice, and lemon zest in a blender or food processor and blend to a very smooth texture, about 2 minutes. Add the mayonnaise and process to blend thoroughly. Add 1 tablespoon of the reserved cooking broth. Add more, if desired, to taste or for a thinner consistency. Refrigerate in a tightly sealed container for at least 2 and up to 24 hours. (The sauce tastes best if made 1 day in advance and allowed to rest overnight).
To serve, thinly slice the meat and arrange on a platter. Drizzle the sauce over each slice, season generously with the kosher salt and black pepper. Garnish by sprinkling with additional capers and chopped parsley. Arrange the lemon slices around the platter and serve.