In a large bowl of stand mixer, combine the brown sugar and 2 cups of warm water until dissolved. Sprinkle the yeast over the water and let stand until foamy, about 5 minutes. Stir in the vegetable oil and 3 cups of the flour. Knead in the remaining 2¾ cups of 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 ¼ cup more flour.

Transfer the dough to a large, greased bowl, cover with plastic wrap and let stand at room temperature until doubled, about 45 minutes.

Preheat the oven to 450°. Line 3 large baking sheets with parchment paper and butter the paper. Punch down the dough and turn it out onto a floured surface. Knead the dough lightly, flatten it out and cut it into 24 equal pieces. Roll each piece into a 7-8 inch stick about ½ inch thick. Transfer the sticks to the prepared baking sheets, leaving at least 2 inches between them. Let stand uncovered until puffed, about 25 minutes.

Heat up 1 cup of hot water on burner. Keep hot until needed. (Or heat in microwave.)

In a large, deep skillet, stir the baking soda into 2 quarts of water and bring to a simmer over high heat. Reduce the heat to medium. Using 2 slotted spoons, carefully transfer 6 pretzel sticks at a time to the simmering water for a total of 30 seconds. Turning once after 15 seconds. Add about 1 cup of hot water before cooking the second batch of pretzels. Transfer the pretzel sticks to paper towels to drain, then return them to the baking sheets, spacing them evenly. Add another cup of hot water before the second batch.

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.