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In a large, heavy-bottomed skillet, heat the butter over medium heat. Add the chopped zucchini flesh, carrot and celery and cook, stirring occasionally, over medium heat until the vegetables are softened and slightly browned, about 8 to 9 minutes. Season with salt and pepper and add the remaining chicken livers-bread crumb mixture. Add the milk and cook, covered, 10 minutes.
Stir in the tomatoes, season with salt and pepper and bring to a boil. Reduce to a simmer, carefully place the zucchini halves in the sauce and cook, covered, 1 hour over medium-low heat. Arrange 2 zucchini halves and some of the sauce on each plate, topping with more Parmigiano-Reggiano, if desired, and serve immediately.