1. To make the polenta: Lightly spray a deep 11-by-15-inch lasagna dish with olive oil spray and set aside. In a large deep saucepan, heat the oil over medium-high heat. Add the onion and sauté for 5 minutes, or until softened. Add the garlic and sauté for 1 minute, or until softened, being sure not to brown it. Add the broth and salt to taste and bring to a rolling boil. In a thin stream, very slowly add the polenta, stirring constantly with a wooden spoon. Lower the heat and cook for about 3 minutes, stirring constantly to be sure it doesn’t stick, until the polenta is very smooth and stiff. Stir in the Parmesan cheese and then quickly pour the polenta into the prepared baking dish; smooth the top with a spatula if necessary. Allow the polenta to sit at room temperature for 2 hours, or until set.

2. To make the meat sauce: In a large Dutch oven, heat 2 tablespoons of the olive oil over medium heat. Add the sirloin and veal and cook, stirring and breaking up the meat, for about 4 to 5 minutes, or until lightly browned. Drain off excess fat, using a colander, and return the meat to the pot. Add the milk and simmer for about 3 to 4 minutes, or until the meat has absorbed the milk. Transfer to a bowl, cover, and set aside.

3. Add the remaining 2 tablespoons oil to the pot and heat over medium heat. Add the onions and carrots and sauté, stirring occasionally, for 4 to 6 minutes, or until softened but not browned. Add the garlic and sauté for 1 minute. Add the wine and boil for 3 to 4 minutes, or until it is reduced by one-third. Add the browned meat, the tomato sauce, basil, and cheese rind, and season with salt and pepper, partially cover, and reduce the heat to medium-low. Simmer gently, stirring occasionally, for about 35 to 40 minutes, or until the sauce has a deep rich flavor and is nicely thickened. Remove from the heat and remove the cheese rind.

4. Meanwhile, make the béchamel sauce: In a medium saucepan, melt the butter over medium heat. Whisk in the flour and cook, whisking, for about 2 minutes; do not allow the flour to color. Add the warm milk and nutmeg and season with salt and pepper, whisking to blend. Cook the sauce for about 5 minutes longer, or until it coats a spoon. Set aside.

5. Preheat the oven to 375°F. Remove the polenta from the baking dish by inverting it onto a work surface; set the dish aside. Cut the polenta crosswise in half, to make 2 squares (this will make it easier to transfer the polenta back into the baking dish later on). The easiest way to slice the polenta into layers is to use a piece of dental floss: Hold a 2-foot-long piece of floss taut between your hands, place the floss against the side of one of the squares farthest from you, and pull the floss toward you through the center of the square, slicing the polenta into two layers. (Alternatively, you can use a serrated knife to slice the polenta.) Repeat with the second square.

6. Lightly spray the lasagna dish with olive oil spray again. Spread 2 cups of the meat sauce evenly over the bottom of the lasagna dish. Place 2 of the polenta squares side by side on top of the sauce. Spoon over another 2 cups of the meat sauce and then pour 1 cup of the béchamel sauce evenly over it. Smooth with a spatula. Sprinkle with ½ cup of the Parmesan cheese. Place the remaining layer of polenta on top, then 2½ cups more of the meat sauce and the remaining béchamel (reserve the remaining meat sauce for another use). Sprinkle with the remaining ½ cup Parmesan cheese. (The lasagna will be slightly above the edge of the dish.)

7. Place the lasagna dish on a baking sheet to catch any drips. Bake, uncovered, for 35 to 40 minutes, or until the cheese is melted and the lasagna is bubbling. Remove from the oven and let sit for 10 minutes.

8. Garnish the lasagna with the parsley. To serve, cut into portions and use a spoon or spatula to scoop them out onto individual dishes.