Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat.
Add the diced chicken breasts and cook until lightly browned and cooked through, about 5-7 minutes. Remove chicken from skillet and set aside.
In the same skillet, add the diced onion and minced garlic. Sauté for 3-4 minutes until the onion is translucent.
Add the diced zucchini, broccoli florets, cauliflower florets, and sliced mushrooms to the skillet. Cook for 5 minutes, stirring occasionally, until vegetables are slightly tender.
Return the cooked chicken to the skillet with the vegetables.
In a separate bowl, whisk together the 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 to combine and bring to a gentle simmer for 2 minutes.
Remove the skillet from heat and stir in the shredded cheddar cheese and grated Parmesan cheese until melted and well combined.
Transfer the mixture to a 9×13-inch casserole dish and spread evenly.
Bake in the preheated oven for 20-25 minutes, until the casserole is bubbly and the top is golden brown.
Remove from the oven and let cool for 5 minutes before serving.