Preheat your oven to 375°F (190°C). Roast the garlic cloves in a small pan with a drizzle of olive oil until fragrant and golden, about 15 minutes. Once cooled, mash the roasted garlic into a smooth paste.
Bring a pot of salted water to a boil and blanch the broccoli florets for 2-3 minutes until just tender but still vibrant green. Drain and set aside.
In a mixing bowl, combine the mayonnaise, mashed roasted garlic, Dijon mustard, and grated Parmesan cheese. Whisk until smooth and well blended.
Fold in the shredded chicken, blanched broccoli, and shredded cheddar cheese into the sauce mixture. Mix gently until everything is evenly coated.
Transfer the mixture into a baking dish, spreading it out evenly with a spatula. Sprinkle additional Parmesan and cheese on top for a golden crust.
Cover the dish with aluminum foil and bake in the preheated oven for 20 minutes. Then, remove the foil and bake for another 10 minutes until bubbly and the cheese is golden brown.
Once baked, carefully remove the dish from the oven and let it rest for a few minutes. The top should appear crispy and cheesy, with bubbling edges.
Slice and serve the cheesy chicken broccoli bake hot, enjoying the contrast of crispy cheese and tender vegetables.
