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For the pretzels:
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1.
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In a large bowl, stir the brown sugar into the warm water until dissolved.
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Sprinkle the yeast over the water and let stand until foamy, about 5 minutes.
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Stir in the vegetable oil and 3 cups of the flour.
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Knead in the remaining flour; the dough will be slightly sticky.
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2.
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Transfer the dough to a floured work surface and knead until silky, about 3 minutes.
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If the dough is very sticky, knead in up to 1/4 cup more flour.
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Transfer the dough to a large, oiled bowl, cover with plastic wrap and let it stand at room temperature until doubled in bulk, about 45 minutes.
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3.
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Preheat the oven to 450 F. Line 3 large cookie sheets with parchment paper and butter the paper.
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Punch down the dough and turn it out onto a floured work surface.
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Knead the dough lightly, flatten it out and cut it into 24 pieces.
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Roll each piece into a 9-inch stick about 1/2 inch thick.
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Transfer the sticks to the prepared cookie sheets, leaving at least 2 inches between them.
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Let them stand uncovered until puffed, about 25 minutes.
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4.
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In a large, deep skillet, stir the baking soda into 2 quarts of water and bring to a simmer over high heat.
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Reduce the heat to moderate.
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Using 2 slotted spoons, carefully transfer 6 pretzel sticks at a time to the simmering water and cook for 30 seconds, turning once.
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Transfer the pretzel sticks to paper towels to drain, then return them to the cookie sheets, spacing them out evenly.
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Continue cooking until youve finished all of the pretzels.
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You will need to add about 1 cup of hot water before cooking each batch of pretzels.
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5.
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Lightly beat the egg with the tap water in a small bowl.
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Brush the pretzel sticks with the egg wash and sprinkle with salt.
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Bake until richly browned, about 10 minutes.
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Serve warm or at room temperature, with mustard.
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For the cheddar beer sauce:
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1.
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In a saucepan over medium heat, melt the butter.
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Add the flour and cook, stirring constantly, for 2 minutes.
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Slowly whisk in the milk and beer, increase the heat to medium-high and bring the mixture to a simmer.
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Add the mustard, Worcestershire and hot sauce and cook, whisking occasionally, until the mixture begins to thicken, 3 to 5 minutes.
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2.
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Working in batches, add the cheese in large handfuls, whisking until each batch has completely melted before adding more.
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Season with salt and pepper.
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Serve cheese sauce warm with German soft pretzels.