Mix yeast, coarse salt, sugar, and warm water in a small bowl, whisking until sugar dissolves. Let stand until foamy, about 5 minutes.
Mix flour and cayenne in a large bowl. Using a pastry cutter or your fingers, cut butter into flour until mixture resembles coarse crumbs.
Slowly pour yeast mixture over flour mixture, stirring to combine. Using your hands, gather dough together. Turn dough out onto a lightly floured surface, and knead until it is no longer sticky, about 5 minutes. Cover with plastic, and let rise for 30 minutes.
Cut dough into 12 pieces, and roll each into an 18-inch rope. Form a U shape with 1 rope, and twist ends together twice. Fold twisted portion backward along center of U shape to form a circle, then gently press ends of rope onto dough to seal. Transfer to an oiled baking sheet, and repeat. Let rise for 20 minutes.
Preheat oven to 475 degrees. Bring a large pot of water to a boil, and add baking soda. Boil pretzels in batches until puffed and slightly shiny, 1 to 2 minutes per side. Transfer to wire racks to drain. Return to baking sheet, sprinkle with pretzel salt, and bake until golden brown and cooked through, about 15 minutes. Pretzels will keep, uncovered, at room temperature for up to 12 hours. Rewarm in a 250-degree oven if desired, and serve with mustards.
To save time, store-bought bread dough can be used in place of homemade; start at step 4.