Stir the warm water, sugar, and active dry yeast together. Let the mixture stand for 5 to 10 minutes, until foamy; instant yeast may be mixed directly into the flour instead.

Add the flour, salt, and melted, slightly cooled butter to the yeast mixture. Stir until a rough dough forms.

Knead the dough by hand for approximately 5 to 7 minutes, or with a dough hook for 4 to 5 minutes. Stop when the dough is smooth and elastic, slightly tacky, and not wet.

Place the dough in a lightly greased bowl, cover it, and let it rise in a warm place for 30 to 60 minutes, until noticeably puffy.

Heat the oven to 425°F and line two baking sheets with parchment paper. Divide the dough into 4 to 6 portions, roll each into a rope about 1/2 to 3/4 inch thick, and cut the ropes into pieces approximately 1 inch long.

Bring 10 cups of water to a boil and carefully stir in 2/3 cup baking soda. Boil the dough pieces in batches for 20 to 30 seconds, then remove them with a slotted spoon and drain briefly.

Place the boiled pieces on the prepared baking sheets. Whisk the egg with 1 tablespoon water and brush the pretzel pieces lightly with the egg wash.

Bake for 12 to 15 minutes, rotating the pans halfway through if necessary, until the bites are puffed and deep golden brown.

Combine the 1/2 cup sugar and cinnamon in a bowl. While the pretzel bites are still warm, brush them with the melted butter and toss them with the cinnamon-sugar mixture.