Preheat the oven to 375°F (190°C).
Heat olive oil in a medium skillet over medium heat. Add the chopped onion and cook for 3 minutes until softened.
Add the minced garlic and sliced mushrooms to the skillet. Cook for 4 minutes until mushrooms are tender.
Add the bite-sized chicken pieces to the skillet. Cook for 5-7 minutes, stirring occasionally, until the chicken is no longer pink.
Stir in the diced carrots and frozen peas. Cook for 2 minutes.
Pour in the chicken broth and heavy cream. Add dried thyme, salt, and black pepper. Stir well and let the mixture simmer for 5 minutes until slightly thickened.
Transfer the chicken mixture to a small casserole dish (about 1-quart size). Sprinkle the shredded cheddar cheese evenly over the top.
In a small bowl, combine the breadcrumbs and melted butter. Sprinkle this mixture evenly over the cheese layer.
Bake in the preheated oven for 20-25 minutes, or until the topping is golden brown and the casserole is bubbly.
Remove from the oven and let it cool for 5 minutes before serving.