In a large bowl or the bowl of an electric mixer, place 2 1/4 cups flour, 1/2 teaspoon salt, 1 tablespoon sugar, and 2 1/4 teaspoons yeast . Mix to just combine. Add 8 ounces very warm water and mix well, adding remaining flour, as needed, a bit at a time to form a soft, smooth dough that clears the sides and bottom of the bowl.
Knead the dough, by hand or machine, for about 5 minutes, until it is soft, smooth and quite slack. Lightly oil the dough and place it in a bowl, covered with plastic wrap. Let it rise for 40-60 minutes.
Preheat your oven to 475 degrees. Prepare two baking sheets by lining them with parchment paper or lightly greasing them.
Transfer the dough to a lightly greased work surface, and divide it into about four equal pieces. Roll each piece into a rope, about 1 inch in diameter. Allow the pieces to rest, uncovered, for 5 minutes. Cut each strip of dough into pieces, 1 to 1 1/2 inches in long.
While the dough is resting, bring 12 cups water and 1/2 cup baking soda to a boil in a large pot or Dutch oven. Stir until the baking soda is dissolved. Drop the pretzel bites (about 15 at a time) into the boiling water and let boil for one minute.
Remove the dough from the boiling water with a metal strainer or slotted spoon, allowing all of the water to drain back into the pot. (Instead of boiling, you can dip the pretzel bites in a mixture of 1 cup warm water and 2 tablespoons of baking soda). Allow pretzel bites to rest to rest, uncovered, for 10 minutes. Brush with beaten egg.
Bake one sheet at a time for 5-7 minutes or until the pretzels are golden brown.
Remove the pretzels from the oven, and brush them thoroughly with 3 tablespoons butter, melted. If desired, sprinkle with coarse salt. Pretzels can be reheated in the oven or microwave on low heat.
To make the honey mustard sauce, whisk together 1/2 cup mayonnaise, 4 tablespoons honey, 4 tablespoons grainy Dijon mustard and 1 teaspoon lemon juice in a medium bowl. Serve with pretzel bites.