Sprinkle the chicken with salt, pepper, Italian seasoning, and dried thyme. Place the chicken in a large ziplock bag and drizzle with approximately 1 tablespoon of olive oil. Mix chicken thoroughly to evenly coat with oil and sprinkle with 2-3 tablespoons of all-purpose flour. Seal the bag and toss around the chicken breasts until each is completely coated with flour. Set aside.
Melt 1-2 tablespoons of butter in a large Dutch oven over medium-high heat. Carefully add chicken breasts to the Dutch oven in a single layer. Cook, undisturbed, for 3-4 minutes, before flipping over to cook the other side for an additional 3-4 minutes or until lightly browned and cooked through. Remove to a plate and set aside. Repeat with any uncooked chicken breasts.
To the same pot melt remaining 1 tablespoon butter or olive oil. Add the chopped butternut squash and stir to coat. Cook for 5 minutes, or until squash just starts to soften. Add the chopped onion and cook for 5 minutes, stirring often. Sprinkle with salt and pepper. Add the minced garlic and sauté for 1 minute, stirring continuously.
Stir in the mushrooms with the onion and butternut squash. Sauté the mushrooms for 3-4 minutes, or until mushrooms just start to release their water.
Add chicken back to the pot, carefully nestling among the vegetables and gently stir in the vegetable or chicken broth. Bring liquid to a low boil and cook, uncovered for 1-2 minutes. Reduce heat to low, cover, and cook for an additional 2-3 minutes, gently stirring to prevent burning.
Ladle a few tablespoons of broth into a bowl. Slowly add 3-4 tablespoons of milk, cream or half-and-half and mix. Add in the remaining cream and mix. Slowly pour the bowl of cream into the pot with the chicken and gently mix to evenly distribute the broth and cream. Sprinkle with fresh chopped parsley. Enjoy!