Preheat the oven to 350 degrees F (175 degrees C).
Bring a large pot of water to a boil over medium-high heat and add chicken breasts; reduce heat and let simmer until chicken falls apart easily with a fork and is no longer pink, 10 to 15 minutes. Drain water, shred chicken with 2 forks, and set aside.
While chicken is boiling, bring a large pot of lightly salted water to a boil. Cook lasagna noodles in the boiling water, stirring occasionally, until tender yet firm to the bite, about 8 minutes. Drain and lay noodles in a single, flat layer on a paper towel to dry.
Heat butter in a small saute pan over medium-high heat. Saute onion and garlic in the hot butter just until onion is translucent, 5 to 7 minutes.
Mix shredded chicken, onion-garlic mixture, 1 cup spaghetti sauce, ricotta cheese, 1/2 of the Parmesan cheese, and Italian seasoning together in a large bowl.
Mix remaining spaghetti sauce and remaining Parmesan cheese together in another bowl. Spread a thin layer on the bottom of a glass 9×13-inch baking dish. Layer 3 lasagna noodles, spread chicken mixture on top, then add another thin layer of the sauce mixture. Repeat, leaving a final layer of noodles on top. Top with remaining sauce. Sprinkle mozzarella and Cheddar cheese on top.
Bake in the preheated oven until heated through and cheese bubbles, about 45 minutes.