Preheat the oven to 400°F.
In a small baking dish, toss the garlic cloves with the olive oil and a pinch of salt and pepper. Roast until fragrant and browned, about 20 minutes.
Transfer half of the roasted garlic to a medium pot along with the milk, parmesan, cream, 1/2 teaspoon salt and a pinch of white pepper. Bring to a simmer and cook for 10-15 minutes, until thickened, stirring often.
Arrange the chicken on a baking sheet, drizzle with olive oil and sprinkle with salt and pepper. Roast until cooked through, 15-20 minutes.
In a blender, combine the remaining roasted garlic, olive oil, red pepper flakes and lemon zest until smooth. Season to taste.
Bring a large pot of salted water to a boil for the broccoli. Prepare an ice bath by filling a large bowl with water and ice. When the water boils, add the broccoli for 2-4 minutes, until bright green and crisp-tender, then immediately use a slotted spoon to transfer the vegetables to the ice bath to stop the cooking process. Remove from the ice bath after 1-2 minutes. Toss the broccoli with the garlic-chile sauce. Season to taste with salt and pepper.
Return the large pot of salted water to a boil. Add the pasta and cook according to package instructions. Drain the pasta, return to the pot, and drizzle with olive oil to prevent sticking.
To serve, toss the pasta with the alfredo sauce, divide between plates, top with chicken breasts and broccoli.