Preheat the oven to 450°F (232°C) and line a large baking sheet with parchment paper.

In a stand mixer fitted with a dough hook, combine warm water, yeast, and sugar. Let stand for 5-10 minutes until frothy, then stir in the salt. Add 3 cups of flour to the mixer and stir to combine. Add minced jalapeños and shredded cheddar cheese. Gradually add remaining flour as needed until the dough forms a soft, slightly tacky ball.

In a large bowl, combine the hot water and baking soda, stirring until dissolved. In a separate small bowl, whisk together the egg and 2 tablespoons of water to create an egg wash; set aside.

On a lightly floured surface, divide the dough into four equal sections. Roll each section into a log and cut each log into three pieces, resulting in 12 equal pieces of dough. Roll each segment into a rope about 18-24 inches long. Fold the rope in half, twist the two segments together, and pinch the ends to seal, tucking them underneath. Place each pretzel twist on the prepared baking sheet.

Dip each pretzel twist in the baking soda water mixture for about 30 seconds, then use a slotted spoon to return them to the baking sheet. Re-twist as needed. Brush each pretzel with the egg wash and sprinkle with coarse salt.

Bake for 8-10 minutes, until the pretzels are golden brown. Allow to cool slightly, then brush with melted butter.