In a 12 to 14-inch saucepan over medium heat, heat the olive oil until hot but not smoking. Add the veal pieces and giblets and cook until evenly browned on all sides, about 10 to 15 minutes. Add the onion, carrot, celery and parsley and cook until the vegetables begin to soften, about 5 minutes. Season to taste with salt and pepper. Add the tomatoes and the wine and break up the tomatoes with the back of a spoon. Adjust the seasoning. Lower the heat to let the mixture simmer gently until the wine evaporates, about 10 minutes. In the meantime, roughly chop the reconstituted mushrooms. Once the ragu has simmered for 10 minutes, add the chopped mushrooms and allow the mixture to continue simmering for another 20 minutes.
Preheat the oven to 400 degrees F.
Slice the cooled polenta 1/2-inch squares. Lightly butter a 9-inch by 9-inch by 3-inch glass or ceramic baking pan. Use 1/3 of the polenta to cover the bottom of the pan with a layer of squares lain side-by-side. Cover with 1/2 of the veal and mushroom ragu, then top with a layer of 1/4 of the Parmigiano-Reggiano. Begin again with another layer of polenta squares, topped with 1/2 of the chicken liver mixture, topped with 1/4 of the Parmigiano-Reggiano. Repeat from the first layer of polenta, ending with Parmigiano-Reggiano atop the chicken liver ragu. Grate some more Parmigiano-Reggiano over the top, if desired.
Place the prepared casserole in the oven and bake until the dish is heated through and the Parmigiano on top is brown and bubbly, about 20 minutes. Serve immediately.
In a 3-quart saucepan, heat water and salt until boiling. Drizzle in polenta in a thin stream, whisking constantly, until all polenta is incorporated and mixture begins to thicken. Switch to a wooden spoon, pull off heat and continue to stir until thick as paste.