To make the dough, in a stand mixer mix together the water, brown sugar, yeast, salt, flour, butter, jalapeños, and cheese until it just comes together. Pour the dough out onto a flour surface and knead for a few minutes until smooth. Place the dough in a lightly oiled bowl, cover, and allow to rise for 1 hour.
After the dough has risen, preheat the oven to 425°F and line one baking sheet with parchment paper and another with paper towels. Place 8 cups of water into a large pot (make sure there is plenty of space above the water) and bring to a boil.
Meanwhile, divide the dough into 3 equal-sized balls. On a lightly floured surface, roll out each ball into an 8-inch circle. With a sharp knife or a pizza cutter, cut each circle into 8 equal-sized wedges, like slicing a pie.
To assemble the pastries, gently lift one of the dough triangles, and starting at the wider end opposite the point, roll the dough around one of the sausages. Place the pastries on the parchment-lined baking sheet and repeat for the remaining sausages.
Once the pastries have been assembled and the pot of water is boiling, add 1/2 cup of baking soda to the pot, turn the heat down to medium, and working in batches, lower the pastries into the hot baking-soda bath with either a mesh strainer or a slotted spoon, and dip into the water for 20 seconds. The water may get foamy and rise to the top. If this happens, just turn off the heat and it should subside.
While removing the pastries, shake off any excess water and place on the paper-towel lined sheet. Repeat until all are done. Note that the dough will be a bit puffy and soggy (this is fine), and sometimes they may come a bit unrolled. If so, you can simply re-roll them.
Place the boiled pigs in jalapeño-cheddar pretzel blankets back on the parchment-lined sheet, sprinkle with coarse salt, and bake for 20 minutes or until a deep brown. Serve warm with mustard or queso on the side for dipping.
Notes
If you want to use this dough for buns or rolls, simply form it into those shapes and proceed with the recipe. It's an excellent pastry even without the miniature hot dogs!