Make the sauce: Heat the olive oil in a large skillet over medium heat; add the chicken and cook until browned, turning, 5 minutes. Add the onion, garlic and tomato paste and cook, stirring, until the onion softens, about 5 minutes. Add the wine and cook until mostly evaporated, about 5 minutes. Stir in 3 cups water, the tomatoes, bay leaves, parsley, 1 teaspoon salt and 1/4 teaspoon pepper; bring to a boil. Reduce the heat to medium low and simmer 50 minutes. Break up the chicken with a spoon. Cook until thickened, 20 more minutes. Remove the bay leaves.
Prepare the lasagna: Preheat the oven to 350 degrees F. Bring a large pot of salted water to a boil. Add the noodles and cook as the label directs. Drain the noodles and drizzle with olive oil; set aside.
Meanwhile, place the kale in a large microwave-safe bowl; add 1/2 cup water and a pinch of salt. Cover with plastic wrap and microwave until tender, 5 minutes. Drain and rinse under cold water; squeeze dry, then return to the bowl. Stir in the eggs, ricotta, half of the parmesan and the parsley. Add 1/2 teaspoon salt and a few grinds of pepper. Toss the mozzarella, asiago and the remaining parmesan in a separate bowl.
Spread 3/4 cup of the sauce in a 9-by-13-inch baking dish. Top with one-third of the noodles, overlapping them slightly. Cover with half of the kale mixture, then 2 more cups sauce. Sprinkle with one-third of the mozzarella mixture. Repeat the layers (noodles, kale mixture, sauce, mozzarella mixture). Top with a final layer of noodles, the remaining sauce and the remaining mozzarella mixture. Set on a baking sheet.
Loosely cover with foil and bake 40 minutes, then uncover and bake until golden and bubbly, about 25 more minutes. Let stand 30 minutes before serving.