01 - Dissolve sugar in warm water in a large bowl. Sprinkle yeast over the surface and let stand for 5 minutes until foamy.
02 - Add melted butter and salt to the yeast mixture. Gradually incorporate flour, mixing until a shaggy dough forms. Knead on a lightly floured surface for 5-7 minutes until smooth and elastic, or use a stand mixer with dough hook.
03 - Place dough in a greased bowl, cover, and let rise in a warm location for approximately 1 hour until doubled in size.
04 - Preheat oven to 450°F and line two baking sheets with parchment paper.
05 - Divide risen dough into 8 equal pieces. Roll each piece into an 18-inch rope, flatten slightly, and place a mozzarella stick in the center. Pinch edges tightly to seal cheese completely. Form each into a classic pretzel shape, ensuring no cheese is exposed.
06 - Bring 8 cups water and baking soda to a boil in a large saucepan, then reduce to a simmer.
07 - Gently dip each pretzel into the baking soda bath for 30 seconds. Remove with a slotted spatula and transfer to prepared baking sheets.
08 - Brush each pretzel with beaten egg and sprinkle generously with coarse salt.
09 - Bake for 12-15 minutes until deep golden brown.
10 - Brush with melted butter immediately after baking for enhanced flavor. Serve warm.