Pat the chicken thighs dry and season with 1 teaspoon salt and a few grinds of pepper. Heat 1 tablespoon of the oil a large skillet over medium-high heat. Add the chicken and cook, turning once, until golden brown on both sides, about 8 minutes total; transfer to a plate (the chicken will not be cooked through).
Add the remaining tablespoon oil and then add the mushrooms and 1/2 teaspoon salt. Cook, stirring occasionally, until the mushrooms release their liquid and turn golden brown along the edges, about 8 minutes. Add the onions and continue to cook until the onions are softened, about 5 minutes.
Sprinkle the flour and garlic over the vegetables and cook, stirring, 1 minute. Add the Marsala and bring to a simmer. Simmer until the Marsala reduces slightly, about 3 minutes. Add the chicken broth and bring to a simmer. Return the chicken and any collected juices to the skillet, cover, reduce the heat to medium low and cook until the chicken is cooked through and tender, about 30 minutes.
Bring a large pot of salted water to a boil.
When the chicken is cooked, transfer the thighs to a cutting board and increase the heat under the sauce to medium high. Boil until slightly thickened, about 6 minutes. Meanwhile, coarsely shred the chicken using 2 forks. Return the shredded chicken to the skillet and keep warm.
Cook the rotini in the boiling water according to the package directions.
Serve the rotini topped with the chicken ragout. Sprinkle with parsley and serve with the grated Parmesan or pecorino.