01 - Dissolve sugar in warm water in a large bowl. Sprinkle yeast over the surface and let stand for 5 minutes until mixture becomes foamy.
02 - Stir melted butter and salt into the yeast mixture. Gradually incorporate flour, mixing until a shaggy dough forms. Knead on a lightly floured surface for 5 to 7 minutes until smooth and elastic, or use a stand mixer with dough hook attachment.
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. Line two baking sheets with parchment paper.
05 - Divide risen dough into 8 equal pieces. Roll each piece into an 18-inch rope and flatten slightly. Place a mozzarella stick in the center and pinch dough edges tightly to seal completely around the cheese. Shape each into a traditional pretzel form, 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 lower each pretzel into the simmering 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 to 15 minutes until deep golden brown in color.
10 - Brush with melted butter immediately after baking for added richness. Serve warm with marinara, ranch, or garlic butter for dipping.