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In another shallow bowl, whisk the egg and milk together. Season with salt and pepper.
In a large hot oven-proof skillet with a lid, render the bacon until crispy, about 6 to 8 minutes. Remove the crispy bacon from the pan and reserve. Lay the rabbit, skin side down in the hot bacon fat and brown the rabbit for 3 to 4 minutes on each side. Remove the rabbit from the pan and set aside. Add the onions to the bacon fat and saute for 2 minutes. Add the mushrooms, shallots, and garlic. Saute for 2 minutes. Season with salt and pepper. Add the thyme and bay leaves. Add the rabbit to the vegetable mixture.
Remove the rabbit pieces from the pan and set aside.
Blend the remaining flour and butter together into a smooth paste (beurre manie). Whisk the paste into hot liquid. Bring the liquid to a simmer and cook for 3 to 4 minutes. Add the rabbit back to the pan and continue to cook for 5 minutes. Stir in the parsley. Season with salt and pepper if needed.