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In a large bowl combine brown sugar and 2 cups water until dissolved. Sprinkle the yeast over the water and let is stand until foamy, about 5 minutes. Stir in the vegetable oil and 3 cups of flour. Knead in the remaining 2 ¾ cups flour; the dough will be slightly sticky. Knead until smooth and silky, about 3 minutes. If the dough is very sticky, knead in up to a ¼ cup more of flour.
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Transfer the dough to a large greased bowl, cover with plastic wrap and let stand at room temperature until doubled, about 45 minutes.
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Line 3 baking sheets with parchment paper and butter the parchment. Punch down the dough and turn it out on a floured surface. Knead the dough lightly, flatten it out and cut it into 24 equal pieces. Roll each piece into 7-8 inch sticks about ½ inch thick. Transfer sticks to prepared baking sheets, leaving at least 2 inches between them (8 per pan). Let stand uncovered until puffed, about 25 minutes.
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Preheat oven to 450 F.
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Heat 1 cup water in a small saucepan.
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In a large deep pot or skillet, stir the baking soda in 2 quarts water and bring to a simmer over high heat. Reduce heat to medium. Using 2 slotted spoons carefully transfer 6 pretzels at a time to the simmering water for a total of 30 seconds. Flipping once after 15 seconds. Add 1 cup hot water to the pot before cooking the second batch. Transfer the pretzel sticks to paper towels to drain, then return them to the baking sheet, spacing them evenly.
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Brush the pretzel sticks with the egg wash and sprinkle with salt. Bake until deeply browned, about 10 minutes. Serve warm or at room temperature.