In the bowl of a stand mixer, combine the warm water, instant yeast, sugar and melted butter. Let stand for 5 minutes.

Add the salt and 3 cups of the flour. Mix on low until the dough is shaggy.

Using the dough hook, knead the dough until it is smooth, elastic and clears the sides of the bowl, 6-8 minutes. Add up to 1/2 cup more flour, one tablespoon at a time, as needed.

Turn the dough onto a floured work surface and knead a few times by hand. Shape the dough into a ball and place in a lightly oiled bowl. Cover with plastic wrap and let rise until dough is doubled in size, about one hour.

Preheat the oven to 450 degrees. Line a large baking sheet with parchment paper.

Divide the dough into 6 portions. Roll and stretch each piece of dough into a rope, about 24 inches long. You may need to let the dough rest occasionally during this process. Twist each rope into a pretzel shape, and lay the pretzels on the prepared baking sheet.

In a large pot, combine the ingredients for the boiling solution. Bring to a boil over high heat, then reduce to a low boil. Gently lower one pretzel into the solution and cook for 20 seconds. Remove the pretzel with a slotted spatula and drain before placing it back on the baking sheet. Repeat with remaining pretzels.

In a small cup, stir together the beaten egg and water. Brush each pretzel with the egg mixture. Sprinkle with coarse salt, to taste.

Bake until golden brown, 12-15 minutes. Cool for 2-3 minutes before brushing lightly with melted butter.