Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat.
Add diced onion and cook for 3 minutes until softened.
Add minced garlic and cook for 1 minute until fragrant.
Add chicken pieces to the skillet and cook until no longer pink, about 5-7 minutes.
Add diced zucchini, broccoli florets, cauliflower florets, and sliced mushrooms to the skillet. Cook for 5 minutes, stirring occasionally.
In a medium bowl, whisk together heavy cream, chicken broth, dried thyme, dried oregano, salt, and black pepper.
Pour the cream mixture over the chicken and vegetables in the skillet. Stir well to combine.
Remove the skillet from heat and stir in 1/2 cup shredded cheddar cheese and all of the Parmesan cheese.
Transfer the mixture to a greased 9×13 inch casserole dish.
Sprinkle the remaining 1/2 cup shredded cheddar cheese evenly on top.
Bake in the preheated oven for 25 minutes, or until the casserole is bubbly and the cheese is melted and golden.
Remove from the oven and let it rest for 5 minutes before serving.