Preheat the oven to 375°F (190°C).
Bring a large pot of salted water to a boil. Add the broccoli florets and cook for 3 minutes until bright green and slightly tender. Drain and immediately plunge into ice water to stop cooking. Drain again and set aside.
In a large skillet, melt the butter over medium heat. Add the chopped onion and cook for 5 minutes until softened.
Add the minced garlic to the skillet and cook for 1 minute until fragrant.
Sprinkle the flour over the onions and garlic, stirring constantly for 2 minutes to form a roux.
Gradually whisk in the milk and chicken broth, continuing to whisk until the sauce thickens, about 5 to 7 minutes.
Stir in salt, black pepper, and dried thyme.
Remove the skillet from heat and stir in 1 cup of the shredded cheddar cheese until melted and smooth.
In a large mixing bowl, combine the cooked chicken, blanched broccoli, and cheese sauce. Mix gently to combine.
Transfer the mixture into a greased 9×13 inch casserole dish and spread evenly.
In a small bowl, mix the breadcrumbs with olive oil until evenly coated.
Sprinkle the remaining 1/2 cup cheddar cheese, Parmesan cheese, and breadcrumb mixture evenly over the casserole.
Bake uncovered for 30 minutes until the top is golden brown and the casserole is bubbly.
Remove from the oven and let stand for 5 minutes before serving.