Brown the Chicken: Heat the olive oil in a large pot over medium heat. Add the chicken pieces and cook until golden on all sides, about 5-7 minutes. Remove from the pot and set aside.
Sauté the Aromatics: In the same pot, add the diced onion and minced garlic. Sauté for 3-4 minutes until softened and fragrant.
Cook the Vegetables: Stir in the carrots and mushrooms, cooking for another 5 minutes.
Thicken the Sauce: Sprinkle the flour over the vegetables, stirring to coat evenly. Cook for 2 minutes to remove the raw flour taste.
Simmer: Gradually pour in the chicken broth while stirring to avoid lumps. Add the thyme and bay leaf, then bring to a gentle simmer.
Combine Everything: Return the chicken to the pot and let it simmer for 10-15 minutes until the chicken is cooked through and the vegetables are tender.
Finish the Dish: Stir in the cream or milk, season with salt and pepper, and let it simmer for an additional 5 minutes until the sauce thickens.
Serve: Remove the bay leaf and discard. Garnish with fresh parsley and serve over rice, mashed potatoes, or crusty bread.