In a large saucepan, heat butter and olive oil over medium-low heat. Add diced chicken, carrot, potatoes, shallots and garlic; cook, stirring, until chicken is nearly cooked through. Add the mushrooms and continue cooking, stirring, until mushrooms are tender. Stir in flour until blended; add chicken broth and thyme. Bring to a simmer, stirring. Cover and reduce heat to low; cook for about 15 to 20 minutes, until vegetables are tender. Add salt and pepper to taste; stir in half and half and heat through.