In the bowl of a stand mixer, combine the warm water, yeast, sugar, and salt. Let stand for 10 minutes until foamy.
Add the chopped rosemary, flour, and softened butter.
Knead with a dough hook for about 10–12 minutes until smooth and elastic.
Cover with a damp towel and allow the dough to rise in a warm place for about 1 hour or until doubled in size.
Mix together the mozzarella and Parmesan cheeses and place them in the freezer while preparing the dough.
Preheat the oven to 425°F (220°C) and line two baking sheets with parchment paper.
Bring the water and baking soda to a gentle boil in a large pot.
Divide the dough into 8 equal portions.
Roll each portion into a long rope about 16 inches long, then gently flatten it into a strip about 4 inches wide.
Spoon about 2 tablespoons of the chilled cheese mixture along the center.
Roll the dough tightly around the filling and pinch the seams securely closed.
Shape each rope into a classic pretzel.
Carefully boil each pretzel in the baking soda bath for 30 seconds.
Remove with a slotted spoon and place on the prepared baking sheets.
Brush each pretzel with the egg wash.
Bake the Mozzarella Stuffed Rosemary and Parmesan Soft Pretzels for 16–18 minutes until deeply golden brown.
If desired, brush with melted butter and sprinkle generously with the Parmesan rosemary topping.
Serve warm while the mozzarella is deliciously melted.