In a microwave safe bowl, combine water, salt, sugar, and butter and microwave until butter is melted (about 3 minutes and 30 seconds). Mix butter mixture with a whisk and add to the breadmaker. .
Turn breadmaker on to "dough mixing mode". After dough has been mixed together (about 1 minute after turning on "dough mixing mode") add eggs. Add more flour to mixture if dough is too sticky one tablespoon at a time. .
Allow breadmaker to run through cycle. After cycle is complete, let dough rise in breadmaker until it rises to the top of breadmaker canister (about 1 hour).
Grease a 9 x 4 inch loaf pan with butter.
Dust the surface of the dough with flour. On a floured counter, pat the dough into a 7 by 14 inch rectangle. (If necessary, use a bench scraper to loosen dough and add more flour underneath the dough so it doesn’t stick.). .
Using your bench scraper to cut the loaf in half lengthwise. Gently roll each half into a 14" tube. Pinch the two ends of the dough together and braid dough by laying the left piece over the right until pieces of dough are tightly twisted. Pinch ends together. Transfer loaf to prepared loaf pan. .
Cover pan loosely with plastic wrap and allow to rise until doubled in size. Top of the loaf should rise about 1 inch over the lip of the pan. Could take up to 3 hours. (Tip: Turn oven to 110° then turn off. Let dough rise in warmed oven. Dough will rise in about 1½ hours.) .
Before baking, beat 1 egg to create an egg wash and spread over top of bread. (You will not use entire egg for egg wash.).
Preheat oven to 350º. Bake approximately 25 minutes until the crust is golden brown and internal temperature registers 200º. .
Let cool for 5 minutes in the pan on a wire rack. Remove loaf from pan and cool completely on a wire rack before slicing, about 2 hours.