In the pressure cooker, heat the butter over medium-high heat, being careful not to let it brown. Sauté the pieces of rooster and turn them frequently to brown them evenly, about 10 minutes.
Add the onions, shallots, carrot and garlic. Stir to combine then add the flour and mix again to distribute it well. Once everything is well heated, add the cognac and flambé it (taking all the necessary precautions not to burn yourself).
When there are no more flames, pour in the red wine, water and tomato puree. Season with salt and pepper, add the bouquet garni and close the pressure cooker. Leave to cook under pressure for about 50 minutes. If you are using a chicken rather than a rooster, check the cooking time on your pressure cooker's manual, it will take less, about 30 minutes.
While the rooster is cooking, brown the lardons in a large pan, about 5 minutes. When they nicely browned, add the button mushrooms and sauté for 5 minutes over high heat. Set them aside on a serving platter and keep them warm in the oven.
When the rooster is cooked, remove it from the pressure cooker with a skimmer and place it on the serving dish with the mushrooms and bacon. Remove the bouquet garni from the pressure cooker and boil down the sauce for a few minutes. Adjust the seasoning then pour the sauce over the rooster, sprinkle with fresh rosemary and serve.