Using a small piece of cheesecloth, make a bouquet garni: Wrap 3 garlic cloves, the peppercorns, thyme, bay leaf, and parsley stems; tie in a bundle with kitchen twine. Season chicken with salt and pepper.
In pressure cooker, heat 1 tablespoon olive oil and the butter over medium heat. Brown chicken in batches without crowding, until golden brown, about 3 to 4 minutes. Remove from pan, and set aside.
Pour in the remaining tablespoon olive oil, and place over medium-high heat. Add mushrooms, onions, and remaining 3 cloves garlic, and season with salt and pepper. Cook until golden, about 6 minutes. Remove from pan, and set aside. Return pan to heat.
Add cognac and wine. Using a wooden spoon, scrape up any browned bits from the bottom of the pan. Add chicken stock, tomato paste, bouquet garni, and the reserved chicken, and bring to a boil. Reduce heat to medium, attach cover, and allow pressure to build to the second red ring. Adjust temperature to maintain pressure, and cook for 10 minutes. Turn off heat, and allow pressure to decrease using natural release method (see information booklet).
Add the reserved mushrooms and onions. Attach cover, and place cooker over medium heat. Allow pressure to build to the second red ring. Adjust heat to maintain pressure. Cook for 2 minutes. Repeat pressure reduction using normal method. Transfer the chicken and vegetables to a bowl using a slotted spoon. Discard bouquet garni. In a small bowl, combine cornstarch and water, stirring until cornstarch is dissolved; set aside
Over medium-high heat, cook remaining liquid until reduced by half, 16 to 18 minutes. Whisk in cornstarch mixture, and cook until thickened, about 2 minutes. Return chicken and vegetables to the pot, and stir in parsley. Cook for 2 minutes, and serve garnished with thyme sprigs.