1. Using a small piece of cheesecloth, make a bouquet garni: Wrap 3 garlic cloves, peppercorns, thyme, bay leaf, and parsley stems; tie in a bundle with kitchen twine.

2. Cut chicken into strips about 2 inches long and 3/4 inch wide. In a large, deep skillet or a Dutch oven, heat 1 tablespoon oil over medium-high heat. Add onions and remaining 3 cloves garlic; cook, stirring occasionally, until they begin to brown, about 4 minutes. Add mushrooms; cook until golden, about 4 minutes. Transfer to a dish.

3. Add butter and remaining 1 tablespoon olive oil to skillet. Season chicken with salt and pepper; cook in two batches until browned, about 1 minute per side. Return all chicken to pan. Add cognac and wine. Use a wooden spoon to scrape up any cooked-on bits from the bottom. Add chicken stock; stir in tomato paste. Add bouquet garni. Bring to a boil; reduce heat to a simmer, and cook, covered, 15 minutes. Add the reserved mushrooms, onions, and garlic; cook 5 minutes more.

4. Using a slotted spoon, transfer chicken and vegetables to a bowl. Discard bouquet garni. Over high heat, reduce broth by half, about 12 minutes. In a small bowl, dissolve cornstarch in 1 tablespoon cold water. Pour back into broth, and stir until incorporated. Cook 2 minutes. Return chicken, onions, and mushrooms to pot; cook over medium-low heat until heated through. Chop parsley leaves, and stir in. Garnish with thyme. Serve hot.