Step 1

Preheat the oven to 400˚ F. Butter an 11-by-14-inch or other 4-quart baking dish. Bring a large pot of salted water to a boil.

Step 2

Put the chicken in a dutch oven or other large pot, then add the chicken broth, crushed garlic, lemon zest, rosemary sprigs and bay leaves. Partially cover and bring to a simmer over medium-low heat. Cook, partially covered, until the chicken is tender, 20 to 22 minutes. Strain through a fine-mesh sieve (you should have 2 cups liquid; if necessary, add a little water to make 2 cups). When the chicken is cool enough to handle, tear or shred into small pieces. Wipe the dutch oven clean.

Step 3

Meanwhile, add the pasta to the boiling water and cook until just al dente, adding the broccoli during the last 2 minutes of cooking. Drain the pasta and broccoli in a colander and rinse with cold water to stop the cooking. Let drain in the colander while you make the sauce.

Step 4

Melt the butter in the Dutch oven over medium heat. Add the chopped garlic and cook until sizzling, about 30 seconds. Sprinkle in the flour and stir to make a paste; cook until the flour is no longer raw, 1 to 2 minutes. Whisk in the reserved 2 cups cooking liquid and the heavy cream; season with 1 teaspoon salt and a few grinds of pepper. Return to a simmer, whisking to make a smooth sauce. Cook over low heat until thickened, 3 to 4 minutes.

Step 5

Remove the pot from the heat. Stir in 1 cup Parmesan, the parsley and lemon juice. Add the chicken, pasta and broccoli and stir to coat. Transfer to the baking dish and sprinkle with the remaining 1/4 cup Parmesan and the mozzarella. Bake until the top is golden and the edges are bubbly, about 25 minutes. Let rest 5 minutes before serving.