Combine the yeast, milk, and honey in the bowl of an electric mixer fitted with the dough hook. Let sit for 10 minutes, until frothy.
Stir in the melted butter and salt. Add the flour, and mix on low until a soft ball forms, about 3 minutes, scraping down the sides of the bowl as necessary. If the dough feels very sticky, add one more tablespoon of flour.
Transfer the dough ball to a large lightly-oiled bowl, then turn the ball so all sides are coated. Cover with plastic wrap and set in a warm place. Let rise until doubled in size, 1 to 1 1/2 hours. Punch down the dough, re-cover, and let rise for another 30 minutes.
Line two large baking sheets with parchment paper. Remove the dough from the bowl and cut into 6 or 8 pieces, depending on how large you want your pretzels. Starting at the center and rolling outwards, use the palm of your hands to roll each piece into a 20-inch long rope. Set the rope on a clean work surface, form a 'U' shape, and twist the ends of the rope around each other twice. Then, attach the ends to the bottom of the 'U' to form a pretzel shape.
Heat the oven to 450 °F. In a wide saucepan or Dutch oven, bring 6 cups of water and baking soda to a boil. Using a spatula, carefully slide two pretzels into the water. Let cook for 30 seconds, then use the spatula or a set of tongs to flip the pretzels over. Boil 30 more seconds, remove them from the water using a slotted spoon, draining any excess water as you go. As you place the boiled pretzels onto the prepared baking sheet, flip them so that they are right side up. Repeat with the remaining pretzels.
Whisk the egg with one teaspoon water in a small bowl. Brush the pretzels with the egg mixture, then sprinkle with coarse salt. Bake until golden brown, 11-12 minutes.