Season chicken thighs with salt and pepper, to taste.
Heat olive oil in a large stockpot or Dutch oven over medium heat. Add chicken to the stockpot and cook until golden, about 2-3 minutes; set aside.
Melt butter in the stockpot or Dutch oven over medium heat. Add mushrooms, onion, carrots and celery. Cook, stirring occasionally, until tender, about 3-4 minutes.
Stir in garlic and thyme until fragrant, about 1 minute.
Whisk in flour until lightly browned, about 1 minute.
Stir in chicken stock, bay leaf and chicken, and cook, stirring constantly, until slightly thickened, about 4-5 minutes.
Stir in half and half until heated through, about 1-2 minutes; season with salt and pepper, to taste. If the soup is too thick, add more half and half as needed until desired consistency is reached.
Serve immediately, garnished with parsley, if desired.
Notes
*Half and half is equal parts of whole milk and cream. For 1 cup half and half, you can substitute 3/4 cup whole milk + 1/4 cup heavy cream or 2/3 cup skim or low-fat milk + 1/3 cup heavy cream.