How to Make It
In a large saucepan, combine the lentils, water, onion, garlic, and 1 1/2 teaspoons of the salt. Bring to a boil, reduce the heat, and simmer, partially covered, until the lentils are just tender but not falling apart, about 35 minutes. Drain and transfer to a large bowl to cool.
In a large glass or stainless-steel bowl, whisk together the vinegar, mustard, 3/4 teaspoon of the salt, and 1/4 teaspoon of the pepper. Add the 1/2 cup oil slowly, whisking. Add 1/2 cup of this vinaigrette to the lentil salad along with the tomatoes and parsley.
In a shallow dish, combine the cornmeal, Parmesan, and the remaining 1/2 teaspoon salt and 1/4 teaspoon pepper. Put the eggs in another shallow dish. In a large nonstick frying pan, heat 1 tablespoon of the oil and 1 tablespoon of the butter over moderate heat. Dip half the livers in the egg and then into the cornmeal mixture; fry for 2 minutes. Turn and cook until browned but still pink inside, about 2 minutes longer. Repeat with the remaining livers and the remaining 1 tablespoon each oil and butter.
To serve, toss the greens in the remaining vinaigrette and put on plates. Top with the lentils and livers.