Season chicken with salt and black pepper.
Heat 1 tbsp olive oil in a large pot over medium-high heat. Add chicken and cook for 3-4 minutes, until lightly browned. Remove and set aside.
Add remaining 2 tbsp olive oil to the pot. Add mushrooms, onion, carrots, and celery. Cook over medium heat for 5-6 minutes, stirring occasionally, until softened.
Add garlic, oregano, and thyme. Cook for 1 minute, until fragrant.
Add chicken stock, bay leaf, and chicken. Bring to a simmer, then reduce heat to medium-low. Cover and cook for 10-15 minutes, until the chicken is cooked through and carrots are tender.
In a small bowl, whisk cornstarch with 4 tbsp cold water until completely smooth.
Slowly stir the cornstarch mixture into the simmering soup. Cook for 2-3 minutes, stirring, until slightly thickened.
Reduce heat to low and stir in half and half. Cook for 2-3 minutes, until warmed through. Do not boil.
Remove bay leaf. Taste and adjust salt and pepper as needed.
Serve warm, topped with fresh parsley.