In a small pot or microwave safe bowl, heat the milk and butter together on the stove top or in a microwave oven. When the butter melts, set the mixture aside and allow it to cool to 110-115°F. The mixture should feel warm (not hot!) to the touch.
Meanwhile, in a quart-size glass measure, whisk together the water, yeast, and sugar. When the milk mixture has cooled sufficiently, add it and the eggs to the yeast mixture. Whisk briefly to combine.
Tip 3 1/2 cups flour into a large mixing bowl. Add the salt, and stir to mix. Make a well in the center of the flour, and then add the liquid ingredients. Stir vigorously, with a stout spoon until a thick, smooth batter develops. If the batter seem soupy rather than thick, go ahead and stir in the extra 1/2 cup of flour. Cover the bowl with cling film and let rise at room temperature until doubled in volume -- about 2 hours.
Use a spoon or spatula to gently deflate the batter. Then pour the batter into a 9x5 loaf pan or a 13-inch pain de mie ("Pullman") pan. Cover loosely with lightly-greased cling film. If using a loaf pan, let the batter rise until it forms a crown approximately 1 inch above the rim of the pan (about 45 minutes at room temperature). If using a pullman pan, let rise just until the batter reaches 1/2 below the rim (about 45 minutes), and then slide the lid on. Halfway through rising time, preheat the oven to 375°F.
Bake in the preheated 375°F until the bread is done -- usually 45 minutes. Unmold and let cool to room temperature on a wire rack before slicing.