Heat the olive oil in a large pot over medium-high heat. Season the chicken with salt and pepper. Working in two batches, add the chicken to the pot and cook until browned, 4 to 5 minutes per side. Remove to a plate. Add the onions to the pot and cook, stirring, until browned, 3 to 5 minutes. Add the garlic and rosemary and cook, stirring, until softened, about 30 seconds. Stir in the wine, scraping up any browned bits, and simmer until reduced by half, 1 to 2 minutes. Return the chicken and its juices to the pot.
Add 3 bay leaves to the pot along with the tomatoes, 3/4 teaspoon salt and a few grinds of pepper. Bring the sauce to a boil and stir, then reduce the heat and simmer, stirring occasionally, until the chicken is tender and the sauce is thickened, 25 to 30 minutes. Let cool.
Meanwhile, bring a large pot of salted water to a boil. Fill a large bowl with ice water. Line a baking sheet with a kitchen towel. Cook the lasagna noodles in the boiling water a few at a time until just al dente, 2 minutes. Carefully remove the noodles with a slotted spoon, transfer to the ice water and let cool, then remove the noodles to the baking sheet and blot dry. Drizzle the noodles very lightly with olive oil to prevent sticking; set aside. Preheat the oven to 375 ̊ F.
Make the béchamel: Melt the butter in a large saucepan over medium heat. Whisk in the flour; cook, whisking constantly, until smooth, 1 to 2 minutes. Increase the heat to medium high and whisk in 1 cup milk until combined. Continue whisking in the remaining 3 cups milk, 1 cup at a time, until combined. Add the remaining bay leaf, a big pinch each of salt and pepper and the nutmeg and cook, whisking, until the sauce comes to a simmer. Reduce the heat to maintain a simmer and cook until thickened, 5 to 7 minutes. Season with salt and pepper and let cool slightly, about 10 minutes. Discard the bay leaf.
Butter a 9-by-13-inch baking dish. Shred the chicken in the sauce with two forks; discard the bay leaves. Spread 1 1/2 cups of the chicken sauce in the baking dish and top with a layer of noodles, overlapping them slightly and trimming if necessary. Top with another layer of the chicken sauce, 1/2 cup béchamel and 1/4 cup Parmesan. Repeat the layering three more times (noodles, chicken sauce, béchamel, Parmesan), then top with a final layer of noodles (you might not use all of the noodles). Top with the remaining béchamel and parmesan. Put the lasagna on a rimmed baking sheet.
Bake the lasagna until bubbling around the edges and lightly browned on top, 30 to 35 minutes. Let sit 15 minutes before serving. Top with more Parmesan.