In a medium-sized bowl, stir together the flour, yeast, sugar, and salt. Gradually add the warm milk, butter, and 2 eggs into the flour mixture; knead until the dough is smooth. The dough is ready to rise when it is completely smooth and pulls away from the sides of the bowl. Cover the bowl, and allow the dough to rise for 2 hours, or until it is doubled in size.
Transfer the dough from the bowl onto a floured work surface and punch it down a few times. Divide the dough into 3 equal-sized balls. Roll each ball into a 10-inch long rope, and then braid the ropes together. Tuck the ends under and place the braid in a greased 9-inch by 5-inch baking pan. Alternately, you can punch the dough down and form it into a simple oval. Cover the pan and allow the dough to rise for an additional 45 minutes to 1 hour, until it is doubled in size.
Preheat the oven to 400 degrees. Remove the dough covering, gently brush the loaf with the beaten egg, sprinkle with a small amount of sugar, and bake for 10 minutes. Reduce the heat to 350 degrees and bake for an additional 25 minutes, until the brioche is golden brown. Allow it to cool for 5 minutes in the pan, and then transfer it to a wire cooling rack.
Makes 10 servings.