Preheat your oven to 350°F (175°C).
In a large skillet, heat the olive oil over medium heat. Add the diced onion and garlic, cooking until softened, about 3-5 minutes.
Add the broccoli and carrots to the skillet, sautéing for another 5-7 minutes until slightly tender. Remove from heat and set aside.
In a mixing bowl, combine the cream of chicken soup, sour cream, chicken broth, paprika, salt, pepper, and thyme. Whisk until smooth.
In the baking dish, arrange the chicken breasts in a single layer. Pour the vegetable mixture over the chicken.
Pour the creamy sauce over the chicken and vegetables, ensuring they are evenly coated.
If using, sprinkle breadcrumbs and shredded cheddar cheese on top of the casserole for added texture and flavor.
Cover the dish with aluminum foil and bake in the preheated oven for 25 minutes.
Remove the foil and bake for an additional 15 minutes or until the chicken is cooked through and the casserole is bubbly.
Let it cool for a few minutes before serving.