Melt 4 tablespoons of butter in a large skillet or Dutch oven over medium-low heat. Add the sliced yellow onions and cook, stirring occasionally, for 15 to 20 minutes, until deeply golden brown and caramelized.

Increase the heat to medium and pour in the 1/2 cup of beef broth and the Worcestershire sauce, scraping the bottom of the pan to lift any browned bits. Stir in the dried thyme. Set the onions aside.

Preheat oven to 375°F (190°C). Lightly grease a 9x13 inch casserole dish. Place the drained broccoli florets into the prepared casserole dish.

In a separate medium saucepan, melt the remaining 4 tablespoons of butter over medium heat. Whisk in the flour and cook for 1 minute, stirring constantly, until a smooth paste (roux) forms. Gradually whisk in the 2 cups of whole milk until the mixture is smooth.

Bring the milk mixture to a gentle simmer, whisking frequently. Once the sauce has thickened slightly, reduce the heat to low. Stir in the remaining 1 cup of beef broth, salt, and black pepper. Remove the pan from the heat.

Stir in 1 cup of the Gruyère cheese and all of the sharp white cheddar cheese until completely melted and the sauce is smooth. Pour the prepared caramelized French onions and their liquid into the cheese sauce. Stir well to combine.

Pour the entire cheese and onion mixture over the broccoli in the casserole dish. Gently stir the mixture to ensure the broccoli is evenly coated.

Sprinkle the remaining 1/2 cup of Gruyère cheese evenly over the top of the casserole. Scatter half (about 3 ounces) of the French Fried Onions over the cheese layer.

Bake for 20 to 25 minutes, or until the casserole is bubbly and the cheese is melted.

Remove the casserole from the oven. Increase the oven temperature to 400°F (200°C) and scatter the remaining French Fried Onions over the top. Return the casserole to the oven and bake for an additional 3 to 5 minutes, or until the topping is golden brown and crisp.

Allow the casserole to rest for 5 to 10 minutes before serving.