In a heavy skillet, over medium-low heat, melt 2 tablespoons of butter and saute the onion and mushrooms until the onion is translucent. Stir in the lemon rind, bread crumbs, parsley, 1 teaspoon salt, and 1/4 teaspoon pepper. When the parsley has wilted, about 1 minute after you add it to the skillet, quickly stir in the eggs to bind the mixture and remove skillet from the heat at once. Set aside.
With a rolling pin or a wooden mallet, flatten the beef pieces until each is very thin and about twice its original size. As you finish each piece, season it with a little salt, pepper, and a pinch of thyme. At the widest end of each beef slice, place 1 teaspoon of bread-crumb mixture from the skillet. Roll up the meat, sausage-shape, and secure it with a wooden toothpick through the center. Roll each piece in flour.
In a very large skillet, over medium-high heat, melt 4 tablespoons of butter and brown the pieces of beef. As you finish, place them in the slow-cooker. Pour the water into the skillet; scrape up the pan juices and turn the sauce into the cooker. Cover and cook on Low for about 5 hours.
Before serving: About half an hour before serving, mix the garlic and mustard into the sauce around the beef; cover; turn the heat to High, and cook for 30 minutes. If the sauce seems less flavorful than you like, about 5 minutes before serving add a dab of prepared mustard and a little salt. If the sauce seems thin, leave the cover off during the second cooking period.