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Activate the yeast - In a large bowl, combine warm water, yeast, and sugar. Let sit for 5–10 minutes, until foamy.
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Make the dough - Add melted butter, salt, and 4¼ cups flour. Mix until a dough forms. Knead for 5–7 minutes, adding a little extra flour only if the dough is very sticky. It should be soft and slightly tacky, not dry.
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Let it rise - Cover and let rise until doubled, about 45–60 minutes.
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Shape the bites - Turn the dough onto a lightly floured surface. Divide into 6–8 pieces. Roll each piece into a rope about ¾ inch thick, then cut into 1-inch bites.
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Prepare the bath - Bring 10 cups water to a boil. Add baking soda carefully; it will foam up.
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Boil the bites - Boil the pretzel bites in batches for about 20–30 seconds. Remove with a slotted spoon and place on parchment-lined baking sheets.
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Egg wash and salt - Brush with egg wash and sprinkle with coarse salt.
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Bake - Bake at 425°F for 12–15 minutes, until deep golden brown.
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Finish - Brush with melted butter if desired.