Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat. Add the chopped onion and cook for 3-4 minutes until softened.
Add the minced garlic and cook for another 1 minute until fragrant.
Add the sliced mushrooms, diced carrots, and frozen peas to the skillet. Cook for 5-6 minutes until vegetables are tender. Remove the vegetables from the skillet and set aside.
In the same skillet, add the chicken pieces and cook over medium heat until no longer pink, about 6-8 minutes. Remove the chicken and set aside with the vegetables.
In a medium saucepan, melt the butter over medium heat. Whisk in the flour and cook for 1-2 minutes to form a roux, stirring constantly.
Gradually whisk in the whole milk and chicken broth. Continue whisking until the sauce thickens, about 5-7 minutes.
Stir in the dried thyme, dried parsley, salt, black pepper, and shredded cheddar cheese. Stir until the cheese is melted and the sauce is smooth.
Return the cooked chicken and vegetables to the sauce and stir to combine.
Transfer the mixture to a 9×13 inch casserole dish.
In a small bowl, mix the breadcrumbs and grated Parmesan cheese. Sprinkle evenly over the casserole.
Bake uncovered in the preheated oven for 25-30 minutes or until the top is golden brown and the casserole is bubbly.
Remove from the oven and let it rest for 5 minutes before serving.