For the Dough:
In the bowl of a stand mixer fitted with the dough hook attachment, combine the warm water, yeast, sugar, and salt. Let stand for 10 minutes, or until the mixture begins to foam.
Add the chopped rosemary, flour, and softened butter to the yeast mixture. Knead on medium speed for 12 minutes, until the dough is smooth and elastic.
Cover the dough with a warm, damp cloth and place it in a warm area to rise for about an hour, or until the dough has doubled in size.
Assembly and Cooking:
In a small bowl, combine the shredded mozzarella cheese and grated parmesan. Place the bowl in the freezer until needed.
Preheat the oven to 425°F. Line two large baking sheets with parchment paper and set aside.
In a large pot or Dutch oven, bring the water and baking soda to a boil.
Divide the risen dough into 8 equal rounds. Roll each round into a 16-inch rope.
Roll each rope out to a width of about 4 inches.
Remove the cheese mixture from the freezer. Evenly distribute 2 tablespoons of the cheese filling along each rope of dough.
Tightly roll the dough back into a rope, jelly roll style, pinching the edges together to seal the cheese inside.
Shape the dough into a U shape, cross the ends over each other, and pinch the ends onto the bottom of the dough to form a pretzel shape.
Place each pretzel, one at a time, into the boiling cooking liquid. Cook for 30 seconds per pretzel. Remove with a slotted spoon, allowing any excess water to drip back into the pot before transferring to the prepared baking sheet.
Once all the pretzels have been boiled, brush the tops with the egg wash.
Place the baking sheet in the oven and bake for 16-18 minutes, or until the pretzels are golden brown.
Allow the pretzels to cool on the baking sheet for 5 minutes before handling.
Optional Topping:
In a small bowl, combine the grated parmesan, chopped rosemary, garlic powder, black pepper, and salt.
Brush the top of each pretzel with melted butter, then sprinkle with the topping mixture. Serve immediately.