Combine 2 cups of warm water (110 to 115 degrees F.), the butter and brown sugar in the bowl of stand mixer fitted with a dough hook. Sprinkle the yeast over the water and let stand until foamy, about 10 minutes. Whisk together the flour and kosher salt.
Add half of the flour mixture to the yeast and begin to mix on low speed. Increase the speed to medium as the flour is incorporated (scrape down the sides of the bowl as needed). Lower the speed to low and add the remaining flour. Increase the speed to medium again and continue to beat until the dough gathers around the hook and pulls away from the side of the bowl, about 3 minutes.
Turn the dough out onto a flour-dusted surface and knead by hand until smooth, about 5 minutes, using more flour as needed. Butter the inside of a large bowl; add the dough and turn it to coat all sides with butter. Cover the bowl and let the dough rise in a warm spot until doubled in size, about 1 hour.
Meanwhile, combine the baking soda and 1 cup water in a small saucepan. Bring to a simmer and cook until all of the baking soda is dissolved. Set aside.
Make a slit down the length of each kielbasa link, cutting about halfway through, and stuff each with 1 tablespoon of the shredded cheese.
Position oven racks in the upper and lower thirds of the oven and preheat to 425 degrees F. Generously spray two baking sheets with cooking spray. Punch down the dough and cut it into 12 equal pieces; cover with a towel.
Roll a piece of dough out into an 18-inch rope. Wrap the rope around a kielbasa link, leaving the ends of the kielbasa slightly exposed. Repeat with the remaining dough and kielbasa. Arrange on the prepared baking sheets. Brush the top and sides of each with the baking soda water and sprinkle each with coarse salt.
Bake until the dough is golden brown and the cheese is melted, 25 to 30 minutes. Serve with mustard.