In the pressure cooker fry the bacon until crisp. Use a slotted spoon to transfer the cooked bacon to a bowl and set aside. Add the 1 tablespoon oil to the pressure cooker and bring to a hot temperature.
Season with salt and pepper the chicken pieces. To the pot add the pieces of cut-up chicken and brown in batches in the bacon fat and oil. Transfer cooked pieces to the bowl with the bacon once it's lightly browned.
Add the carrot and celery to the pressure cooker; saute for 2 minutes. Add the onions; saute for 3 minutes or until they begin to brown. Add the garlic, mushrooms, thyme, and parsley; saute for 5 minutes or until the mushrooms have released their moisture.
Stir in the wine and cognac. Return the bacon and chicken pieces. Lock the lid into place and bring to low pressure; maintain pressure for 10 minutes. Remove from heat, quick-release the pressure and remove the lid.
In a bowl, whisk the cornstarch into the water. Return the pressure cooker to medium heat. Bring to a simmer and whisk in the cornstarch slurry.
Simmer for 3 minutes or until the sauce is thick and glossy and the cornstarch flavor has cooked out of the sauce. Adjust seasoning and add salt and pepper to taste.