01 - Dissolve granulated sugar in warm water in a large bowl. Sprinkle active dry yeast over the surface and let stand for 5 minutes until the mixture becomes foamy and fragrant.
02 - Add melted butter and fine sea salt to 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 fitted with a dough hook.
03 - Place the dough in a greased bowl, cover with plastic wrap or a clean towel, and let rise in a warm, draft-free area for approximately 1 hour until doubled in volume.
04 - Preheat oven to 450°F. Line two baking sheets with parchment paper and set aside.
05 - Divide risen dough into 8 equal portions. Roll each piece into an 18-inch rope and flatten slightly. Position a mozzarella stick along the center and pinch dough edges firmly to encase the cheese completely, ensuring no filling is exposed. Form each sealed rope into a classic pretzel shape.
06 - Bring 8 cups of water and baking soda to a rolling boil in a large saucepan. Reduce heat to maintain a gentle simmer.
07 - Gently lower each pretzel, one at a time, into the simmering baking soda bath for 30 seconds. Remove with a slotted spatula, drain briefly, and transfer to the prepared baking sheets.
08 - Brush each pretzel generously with beaten egg and sprinkle with coarse salt to taste.
09 - Bake for 12 to 15 minutes until deep golden brown and the cheese inside is melted.
10 - Immediately after removing from the oven, brush with melted butter for enhanced flavor and shine. Serve warm while the cheese remains gooey.