In a large mixing bowl, stir the warm water, sugar, and yeast together. Let it sit for 5–10 minutes until foamy.
Add the flour, salt, rosemary, and parmesan. Stir until it starts to form a dough, then knead for 8–10 minutes until smooth.
Drizzle olive oil in a clean bowl. Place the dough inside, cover it, and let it rise in a warm spot for about an hour.
Preheat oven to 425°F (220°C). Line a baking sheet with parchment paper.
Punch the dough down and divide it into 8 equal pieces. Roll each into a rope, flatten slightly, and wrap around a mozzarella stick, pinching ends to seal.
Bring a large pot of water to a boil and carefully add baking soda.
Boil each pretzel for 30 seconds, remove with slotted spoon, and place on the prepared baking sheet.
Brush each pretzel with egg wash and sprinkle coarse salt on top.
Bake for 18–20 minutes until golden brown. Serve warm for maximum cheese stretch.