Preheat oven to 450°F. Line a large baking sheet with parchment paper.
In a stand mixer fitted with the dough hook, combine warm water, yeast, and sugar. Let stand for 5-10 minutes, or until frothy. Stir in salt.
Add 3 cups of flour; mix until combined. Incorporate the minced jalapeños and cheddar cheese. Gradually add remaining flour until the dough is slightly tacky but not sticky.
Dissolve baking soda in 4 cups of hot water in a large bowl. In a small bowl, whisk together the egg and 2 tablespoons of water to make an egg wash; set aside.
Divide dough into four sections, roll each into a log, and cut each log into three pieces. Roll each piece into an 18-24 inch rope, fold in half, and twist.
Dip each pretzel in the hot baking soda water for 30 seconds, then place on the baking sheet. Brush with egg wash and sprinkle with coarse salt.
Bake for 8-10 minutes until golden brown. Cool slightly and brush with melted butter.