Melt butter in a 6-quart pot over medium heat. Cook onion for 5 minutes, then garlic for 30 seconds.
Stir in flour for 1 minute. Gradually whisk in broth until smooth, then bring to a simmer.
Add broken lasagna noodles, salt and pepper. Simmer uncovered for 12 to 15 minutes, stirring frequently to separate the flat pieces, until noodles are tender.
Stir in milk and chicken and heat gently for 3 to 4 minutes until chicken reaches 165°F. Fold in spinach for 1 minute until wilted.
Switch off the heat and stir in Parmesan. Divide among 6 bowls and add one-sixth of the ricotta to each as a small dollop.