Mix all ingredients together except for the butter in a stand mixer. Mix on 1st speed for about 5 minutes. Turn the mixer on 2nd speed and mix until the dough comes together and pulls away from the side of the bowl. Turn the mixer back to 1st speed and add butter to incorporate. Turn back to 2nd speed and mix until the dough pulls away from the side of the bowl and is shiny and elastic.
Fermenting the Dough
Form the ball into a round shape and cover with plastic wrap. Let ferment all day or overnight at room temperature (about 8-10 hours), so it doubles in size. A sourdough culture ferments dough much more slowly than dry yeast, so you won't have to worry about it over-fermenting.
Shaping and Proofing the Dough
Take the dough out of the bowl and gently flatten to a rectangle shape no wider than the loaf pan you are using to bake it in. Roll the dough towards you, tucking as you go. It doesn't need to be really tight, just secure. Seal the edges with your thumb or heel of your hand.
Butter the loaf pan and cover the bottom (not the sides) with parchment paper and butter that too to keep the bread from sticking.
Place the dough seam side down in the pan and proof in a warm spot in your kitchen. On top of the stove with the stove turned on is a good way to proof the dough. Proof for 90 minutes or up to 2 hours if needed until the dough has expanded by about 30%.
Once the dough is proofed, brush the top with a beaten egg. Sprinkle with coarse sugar or regular sugar. Cut a slit down the middle to "score" the dough so it will rise in the oven. Use a bread lame or small serrated knife.
Baking the Dough
Bake at 350 degrees for 50-60 minutes or until a knife inserted comes out clean. Let cool for a bit before removing from the pan and slicing.