Preheat the oven to 375°F (190°C).
In a small skillet over medium heat, melt the butter.
Add the chopped onion, celery, and carrot to the skillet and sauté for 5 minutes until the vegetables are tender.
Sprinkle the flour over the vegetables and stir to combine. Cook for 1 minute to remove the raw flour taste.
Gradually whisk in the chicken broth and milk, stirring constantly until the sauce thickens, about 2-3 minutes.
Add the salt, black pepper, and dried thyme to the sauce and stir well.
Add the chicken pieces and frozen peas to the skillet. Stir to coat the chicken and peas with the sauce.
Transfer the mixture to a small oven-safe casserole dish (approximately 1-quart capacity).
Sprinkle the shredded cheddar cheese evenly over the top.
Top with the crushed buttery crackers evenly.
Bake in the preheated oven for 25 minutes, or until the casserole is bubbly and the topping is golden brown.
Remove from the oven and let it rest for 5 minutes before serving.