Preheat the oven to 450°F (232°C). Cover a large baking sheet with parchment paper to prevent sticking and promote even baking.
Pour the warm water into the bowl of a stand mixer fitted with a dough hook. Add the active dry yeast and granulated sugar, then leave the mixture undisturbed for 5–10 minutes, or until a foamy layer develops.
Mix in the salt, followed by 3 cups of the all-purpose flour. Continue mixing until the ingredients begin to form a dough.
Add the minced jalapeños and shredded cheddar cheese, mixing until they are evenly distributed.
Gradually incorporate additional flour in small amounts until the dough gathers into a cohesive ball. The finished dough should feel slightly tacky without clinging heavily to the hands or bowl. The full 4 cups of flour may not be required.
Place the very hot water in a large bowl. Stir in the baking soda until fully dissolved.
In a separate small bowl, whisk the egg with 2 tablespoons of water until smooth. Set the egg wash aside.
Transfer the dough to a lightly floured work surface and divide it into four equal portions.
Shape each portion into a log, then cut every log into three pieces to produce 12 evenly sized portions.
Roll each piece into a rope approximately 18–24 inches long. Fold the rope in half and twist the two lengths around one another.
Pinch the ends together and tuck them underneath to secure the twist. Arrange each shaped pretzel on the parchment-lined baking sheet.
Carefully lower each shaped pretzel into the hot baking-soda bath for approximately 30 seconds.
Lift it from the bath with a slotted spoon and return it to the prepared baking sheet. Adjust the twist as necessary to restore its shape.
Brush the surface of each pretzel with the prepared egg wash, then scatter coarse salt over the top.
Bake the pretzel twists for 8–10 minutes, or until their surfaces are evenly golden brown.
Allow the baked twists to rest on the baking sheet for a few minutes. While they are still warm, brush them with the melted unsalted butter.