Stir the warm water, yeast, and sugar together. If using active dry yeast, let the mixture stand for about 5 minutes until foamy; instant yeast can generally be mixed in without proofing.

Add the fine salt, melted butter, and 4 cups of flour. Mix until a shaggy dough forms, then knead for 5–7 minutes, adding the remaining flour a little at a time until the dough is smooth, elastic, and only slightly tacky.

Place the dough in a lightly greased bowl, cover, and let it rise in a warm place for 30–45 minutes, or until noticeably puffy. Divide the dough into 4 portions, roll each into a rope about 3/4 inch thick, and cut into 1-inch pieces.

Bring 10 cups of water to a boil, carefully add 2/3 cup baking soda, and reduce the liquid to a steady simmer. Boil the dough pieces in batches for about 20–30 seconds, turning once if needed, then remove them with a slotted spoon.

Arrange the boiled dough pieces on parchment-lined baking sheets. Brush with the egg wash, sprinkle with coarse or pretzel salt, and bake at 425°F (220°C) for 12–15 minutes, until deeply golden brown.

Brush the hot pretzel bites with melted butter if desired and serve warm with your favorite dipping sauce. Allow any leftovers to cool before storing.