This family-friendly casserole is great for a large gathering. You can make and assemble it a day ahead, just bring it back to room temperature before baking. We like boneless skinless chicken thighs because they cook quickly and stay moist and tender.
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.
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.
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.
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.
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.
Boil the pasta until it’s just al dente. It will soak up the sauce and soften in the oven.