Prepare the following, adding each to the same large bowl as it is completed: Dice 1 medium yellow onion (about 1 ½ cups). Thinly slice 8 ounces white button or cremini mushrooms, halving the mushrooms first if they are larger than 1 ½ inches in diameter. Mince 3 garlic cloves.
Prepare the following, adding each to the same medium bowl: Finely grate 2 ounces Parmesan cheese on the small holes of a box grater (about 1 cup) if needed, or measure out scant ½ cup store-bought grated. Grate 4 ounces low-moisture mozzarella cheese on the large holes of a box grater or measure out 1 cup store-bought shredded.
Melt 4 tablespoons unsalted butter in a large Dutch oven over medium heat. Add the onion mixture, 1 teaspoon of the kosher salt, 1 teaspoon dried Italian seasoning, and ¼ teaspoon black pepper. Cook until the onions are translucent and the mushrooms have released most of their liquid and begin to brown, 12 to 14 minutes.
Meanwhile, butterfly 1 to 1 ½ pounds boneless, skinless chicken breasts: Starting at the thicker end and holding your knife parallel to the cutting board, cut each chicken breast in half. Season all over with the remaining ½ teaspoon kosher salt.
Sprinkle 2 tablespoons all-purpose flour into the pot and cook, stirring constantly, until no dry spots remain, 30 seconds to 1 minute. Slowly pour in 1 (32-ounce) carton low-sodium chicken broth and 1 cup water, stirring to combine and scraping any brown bits from the bottom of the pot. Add the chicken, making sure it is completely submerged, and adjust heat to maintain a simmer.
Cook, stirring occasionally, until the chicken is cooked through and an instant-read thermometer inserted into the thickest part of the chicken registers at least 165°F, 10 to 12 minutes.
Using tongs, transfer the chicken to a clean cutting board. Return the soup to a simmer. Break 8 ounces lasagna noodles into bite-sized pieces and add to the soup. Simmer, stirring occasionally, until al dente, 10 to 14 minutes, or according to package instructions. Meanwhile, use 2 forks to shred the chicken into bite-sized pieces.
Return the chicken to the pot. Add 2 cups baby spinach, ¼ cup heavy cream, and the Parmesan and mozzarella cheeses. Cook, stirring constantly, until the spinach wilts and the cheeses melt, about 1 minute. Taste and season with more kosher salt as needed. Ladle into bowls. Dollop with ricotta cheese and top with more black pepper if desired.