How to Make It
Preheat the oven to 400°F. In a large pot of boiling salted water, cook the pasta until al dente, then drain.
While the pasta is cooking, in a large heavy pot, heat the oil over medium high heat until hot. Whisk in the flour and cook, whisking, until the flour is fragrant, about 3 minutes. Whisk in the milk and 1/2 teaspoon each salt and pepper, then bring to a boil, whisking. Boil until the milk is slightly thickened, about 8 minutes. Remove from the heat and whisk in the cheddar and Monterey Jack cheeses until melted, then whisk in the chicken. Season the sauce with salt and pepper to taste.
Toss the pasta together with the sauce and transfer to a 3-quart baking dish. Sprinkle evenly with the parmesan cheese. Bake the casserole until the filling is bubbling and the cheese is browned on top, 20 to 25 minutes. Serve.