Pat the chicken thighs dry with paper towels and season them all over generously with salt & pepper. Heat the oil in a dutch oven over medium-high heat. Sear the chicken thighs skin-side down until browned, about 5 minutes. Flip the chicken over and cook for 2 more minutes. Transfer the chicken thighs to a plate and set aside leaving any fat or oil in the dutch oven.
Reduce the heat to medium and add the butter and onion to the dutch oven. Cook, stirring often, until the onion is softened, about 5 minutes.
Add the garlic and flour, stir until thoroughly combined. Stir in 4 cups water and 1/2 tsp salt then return the chicken thighs back in the dutch oven and bring to a boil.
When soup begins to boil cover the dutch oven with a tight fitting lid and reduce the heat to medium-low. Simmer over medium-low heat until chicken is cooked through, about 25 minutes.
While the chicken cooks bring a pot of water to a boil. Break up the lasagna noodles into 2-inch pieces and cook them in the boiling water until al dente according to package directions. Drain noodles and set aside.
Transfer the chicken thighs to a plate. Allow chicken to cool for a few minutes then shred the chicken using two forks and discard any skin and bones. Stir the shredded chicken back into the soup.
Stir in the half & half and cook, stirring often, until heated through. Sprinkle in the cheeses and stir until melted and thoroughly combined. Stir in the cooked lasagna noodles and freshly chopped parsley. Season with salt & pepper to taste.