Preheat your oven to 375°F (190°C). If your chicken isn't already cooked, prepare it now. You can boil chicken breasts for 15-20 minutes until cooked through, then shred or dice them. Set aside.

Lightly grease a 9x13 inch baking dish.

In a large skillet or Dutch oven, melt the 4 tablespoons of butter over medium heat.

Add the diced yellow onion and celery to the skillet. Sauté for 5-7 minutes, or until the vegetables are softened and translucent.

Sprinkle the all-purpose flour over the softened vegetables. Cook, stirring constantly, for 1-2 minutes to create a roux. This helps cook out the raw flour taste.

Gradually whisk in the chicken broth, then the whole milk, and finally the heavy cream. Continue whisking to prevent lumps until the sauce is smooth. Bring the mixture to a gentle simmer, stirring frequently, until it thickens, about 5-8 minutes.

Remove the skillet from the heat. Stir in the dried thyme, crushed dried rosemary, salt, and black pepper. Taste and adjust seasonings as needed.

Add the shredded cooked chicken, frozen peas, and frozen diced carrots to the creamy sauce in the skillet. Stir gently to combine everything thoroughly, ensuring the chicken and vegetables are well coated.

Pour the chicken and vegetable mixture evenly into the prepared 9x13 inch baking dish.

In a small bowl, combine the crushed Ritz crackers, shredded sharp cheddar cheese, and 2 tablespoons of melted butter. Stir until the cracker crumbs are evenly coated.

Sprinkle the cracker topping evenly over the chicken mixture in the baking dish.

Bake the casserole in the preheated oven for 30-35 minutes, or until the filling is bubbly around the edges and the cracker topping is golden brown and crispy.

Remove from oven and let stand for 5-10 minutes before serving. This allows the casserole to set slightly and makes for easier serving.