Preheat your oven to 375°F (190°C). In a large oven-safe skillet or Dutch oven, heat the olive oil over medium heat. Add the chopped onion, carrots, and celery. Sauté for 5-7 minutes, or until the vegetables begin to soften. Add the frozen peas and corn and cook for another 2-3 minutes.

Sprinkle the flour over the sautéed vegetables and stir constantly for 1 minute to cook out the raw flour taste. Gradually whisk in the chicken broth and milk, stirring continuously to prevent lumps. Bring the mixture to a simmer, stirring until the sauce thickens. Stir in the dried thyme, crushed rosemary, salt, and black pepper. Add the cooked chicken to the sauce and stir to combine.

Pour the chicken and vegetable mixture into a 9x13 inch baking dish if you are not using an oven-safe skillet. Arrange the refrigerated biscuit dough (or puff pastry, or tater tots) over the top of the chicken mixture. If using biscuits, you can tear them into smaller pieces or leave them whole, depending on your preference.

Bake for 20-25 minutes, or until the topping is golden brown and the filling is bubbly. If the topping starts to brown too quickly, you can loosely cover it with aluminum foil. Once baked, remove from the oven and let it rest for a few minutes before serving.

Serve the Chicken Pot Pie Casserole hot, garnished with fresh chopped parsley if desired. This dish is a complete meal on its own, offering a perfect balance of protein, vegetables, and comforting flavors. Enjoy this simplified classic!