1. Preheat oven to 500 degrees with oven rack in middle position.

2. Bring 4 qt water to boil in a Dutch oven over high heat.

3. Stir in 1 tbsp salt and the pasta and cook, stirring often, until just shy of al dente, about 8 minutes.

4. Reserve 1 1/2 cup pasta cooking water.

5. Drain pasta and toss with olive oil; leave in colander and set aside.

6. Wipe out the Dutch oven with paper towels.

7. Add butter to the pot, return to medium low heat, and heat until melted.

8. Add flour and cook, stirring constantly, until golden, about 1 minute.

9. Stir in garlic and cook until fragrant, about 30 seconds.

10. Slowly whisk in the cream and reserved pasta cooking water.

11. Bring to a simmer and cook, whisking often, until slightly thickened, about 1 minute.

12. Add chicken breasts, partially cover, and bring to a simmer.

13. Reduce heat to low, cover completely, and cook until chicken is done, about 15 minutes.

14. Transfer chicken to a cutting board and remove Dutch oven from heat.

15. Shred chicken into bite-sized pieces, then return to sauce and toss, along with pasta and cheeses, reserving 1/2 cup of the Parmesan.

16. Transfer to a 13x9 baking dish and sprinkle with remaining Parmesan.

17. Bake until top is golden, about 15 minutes.