Step 1In the bowl of a stand mixer or in a large bowl, sprinkle the yeast over the warm water. Stir in the sugar and one-half cup of the bread flour. Set aside until the yeast begins to bubble, about 10 minutes.
Step 2Meanwhile, in a medium bowl, whisk together the remaining bread flour with the rye flour and salt.
Step 3Beat the melted butter into the large bowl with the yeast. Using the dough hook (if using a stand mixer) or a fork or wooden spoon (if mixing by hand), slowly mix in the remaining flour mixture, a spoonful at a time, until all of the flour is added and a firm, thick dough is formed.
Step 4Move the dough to a lightly floured board. Knead the dough until it is smooth and elastic, 2 to 3 minutes.
Step 5Remove the dough to a large, oiled bowl. Cover and set aside in a warm place until the dough is almost doubled in size, 45 minutes to an hour.
Step 6Meanwhile, prepare the pretzel wash and heat the oven to 375 degrees.
Step 7Divide the risen dough into eight pieces, each weighing about 5 ounces. Form each piece into a ball, pinching the seams together at the base of each one. Flatten each ball so it’s about 1 inch thick.
Step 8Coat the pretzels with a wash. If using lye, dip the roll in the wash (wear rubber kitchen gloves and goggles) for 15 to 20 seconds, turning the roll over halfway to coat evenly. Remove the round to a greased nonreactive baking sheet and top as desired (if using an aluminum baking sheet, line the sheet with parchment before greasing). If using beaten egg, brush the egg over the buns.
Step 9Use a serrated knife or razor blade to make a crosswise slit into each roll about one-half-inch deep. Sprinkle over the coarse sea salt. Set the rounds aside until puffed and risen, about 15 minutes depending on the temperature in the room.
Step 10Bake the pretzel rounds, one sheet at a time, in the center of the oven until puffed and a rich golden brown (color will vary depending on the wash), about 20 minutes. Rotate the sheet halfway through baking for even coloring.
Step 11Remove the baking sheet to a rack, and set aside until the pretzel buns have cooled completely before slicing and serving.