Heat butter and milk in a small saucepan over medium heat until butter is melted; let cool to room temperature, about 20 minutes. Whisk together the water and a pinch of the sugar in a small bowl. Sprinkle the yeast over the top of the sugar–water mixture and let stand until foamy, about 10 minutes.
In a large bowl, whisk together 1 1⁄2 cups of the flour, remaining sugar, and salt. Add the cooled milk mixture, yeast mixture, and 2 of the eggs and whisk until smooth, about 2 minutes. Add the remaining flour, 1⁄2 cup at a time, and continue whisking until a dough begins to form, then switch to a rubber spatula and continue mixing. Once all the flour is added, knead the dough in the bowl to form a ball. Turn the dough out onto a very lightly floured surface and knead until smooth, about 5 minutes. Grease a large bowl with butter and add the dough. Turn the dough in the bowl to coat it with butter. Cover the bowl with plastic wrap and let it rise in a warm place until it doubles in size, about 2 hours.
To form dough into the shape known as Saint Joseph's staff, see the Step-By-Step Instructions.
Arrange a rack in the center of the oven; heat to 375°. Beat the remaining egg in a small bowl. Brush the risen dough all over with the egg. Using a razor blade or a thin, sharp knife, make a deep slash down center of staff, about 1" deep; pull the slash apart slightly with your hands. Transfer staff to oven. Using a spray bottle, mist the inside of oven about 10 times; bake bread until deep golden brown, 25–30 minutes. Let bread cool on a rack before serving.