Mix the dough: In a large mixing bowl, whisk together your sourdough starter and water until smooth and milky. Add the flour, sugar, and salt, stirring until a rough dough forms. If using a stand mixer: Knead on medium speed for about 6–8 minutes, until the dough is smooth, elastic, and slightly tacky but not sticky. If kneading by hand: Turn the dough onto a lightly floured surface and knead for 8–10 minutes, until smooth and supple.
First rise (bulk ferment): Place the dough in a lightly oiled bowl, cover with a damp towel or plastic wrap, and let it rise at room temperature for 8–12 hours, or until puffy and well expanded.
Shape the pretzel bites: Gently punch down the dough and divide it into 8 equal pieces. Roll each piece into a rope about 12–15 inches long. Cut each rope into 1-inch pieces. Transfer the pieces to a parchment-lined baking sheet, cover lightly, and let them rise for 1 hour.
Prepare the water bath: In a large pot, bring 8 cups of water to a simmer. Stir in 2 tablespoons of brown sugar and 1 tablespoon of baking soda (add the baking soda slowly to avoid bubbling over).
Boil the pretzel bites: Working in batches, add the pretzel pieces to the simmering water and boil for 15 seconds. Remove with a slotted spoon and place back on the parchment-lined baking sheet. Avoid overcrowding the pot so the pieces cook evenly.
Egg wash and topping: Brush the boiled pretzel bites with a mixture of 1 egg yolk whisked with 1 tablespoon of water. Sprinkle generously with coarse salt.
Bake: Bake at 425°F (218°C) for 12–15 minutes, or until deep golden brown.
Cool & serve: Let the pretzel bites cool slightly before enjoying. They’re delicious plain, dipped in mustard, or with a cheese sauce.