12 hours before making the dough, mix together the sourdough starter, flour, and water until well blended. Cover loosely and let sit for 12 hours.
When ready to make the dough, mix together the starter and water in a large bowl or the bowl or a stand mixer. Add the honey, salt, and bread flour, and mix to combine. Knead the dough for 5-6 minutes by hand or using the stand mixer, until the dough is smooth.
Place the dough in a greased bowl, cover, and let rise for 12 hours.
When ready to bake the pretzels, divide the dough into twelve equal pieces. Roll each piece into a long rope, and shape into a pretzel shape. Place on two greased or parchment lined baking sheets, cover with a towel, and let rest 30 minutes.
While the pretzels are resting, preheat the oven to 425. In a large pot, bring the water, baking soda, and brown sugar to a simmer. Add the pretzels to the pot, 3-4 at a time, cooking for 30 seconds per side, then return the pretzels to the baking sheets. Repeat with the remaining pretzels.
In a small bowl, mix together the egg and water. Brush the egg wash over the pretzels, then sprinkle with the coarse salt.
Bake 12-14 minutes, until golden brown. Remove from the oven and brush with melted butter.