Preheat the oven to 350°F (175°C).
Heat olive oil in a large skillet over medium heat. Add the chicken pieces and cook until no longer pink, about 5-7 minutes. Remove chicken from skillet and set aside.
In the same skillet, add diced onion and cook until translucent, about 3 minutes. Add minced garlic and sliced mushrooms and cook for an additional 3 minutes until mushrooms are tender.
Add frozen peas, carrots, and corn to the skillet and cook for 2 minutes until vegetables are slightly thawed. Remove the vegetable mixture from the skillet and set aside with the chicken.
In the same skillet, melt the unsalted butter over medium heat. Whisk in the all-purpose flour and cook for 1-2 minutes until lightly golden, stirring constantly.
Gradually whisk in the whole milk and chicken broth, continuing to whisk until the sauce thickens and is smooth, about 5-7 minutes.
Season the sauce with salt, black pepper, dried thyme, and dried parsley. Remove from heat.
Return the cooked chicken and vegetable mixture to the skillet with the sauce and stir to combine evenly.
Transfer the mixture into a 9×13 inch baking dish. Sprinkle the shredded cheddar cheese evenly over the top.
Sprinkle the crushed buttery crackers evenly over the cheese layer.
Bake in the preheated oven for 25-30 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.