Fry the bacon slowly in 2 tablespoons of the oil until almost crisp, then transfer to a casserole dish with a slotted spoon. Fry the onions in this fat until soft and translucent. Add the garlic and cook for a further 2 minutes, then add the onions and garlic to the casserole dish.
Season the hare with salt and pepper, and dredge in the flour. Add 2 tablespoons of the oil to the pan, increase the heat to medium and brown the hare pieces on all sides. Transfer to the casserole dish.
Increase the heat under the frying pan, add the wine and deglaze, scraping up any bits clinging to the pan. Pour the wine mixture over the hare ,and add the stock, bouquet garni and salt and pepper. Bring to the boil, skim off any scum that rises to the surface, then reduce the heat, cover and simmer for about 1 hour.
About 15 minutes before the end of the cooking time, fry the pickling onions in the butter and remaining oil. Season lightly with salt and pepper, and sauté for 5–10 minutes until golden brown. Transfer to a dish and keep warm while you fry the button mushrooms. Add these to the onions, and stir both into the casserole dish.
Scatter the parsley over the top, and serve hot.