In a large bowl, combine warm water, yeast, and sugar. Let stand for 5-10 minutes until foamy.
Stir in salt and chopped rosemary.
Gradually add flour and butter, mixing until a shaggy dough forms.
Turn dough onto a lightly floured surface and knead for 6-8 minutes until smooth and elastic.
Place dough in a greased bowl, cover, and let rise in a warm place for about 1 hour, or until doubled in size.
While dough rises, mix shredded mozzarella and grated Parmesan cheese in a small bowl. Divide into 8 equal portions.
Once dough has risen, punch it down and divide it into 8 equal pieces.
On a lightly floured surface, flatten each piece of dough into a rectangle. Place a portion of the cheese mixture in the center.
Fold the dough over the cheese, sealing the edges tightly to fully enclose the filling.
Shape each stuffed dough portion into a traditional pretzel shape.
Whisk together the beaten egg and 1 tablespoon of water in a small bowl for the egg wash.
Place pretzels on a baking sheet lined with parchment paper. Brush generously with egg wash.
Sprinkle coarse salt over the top of each pretzel.
Bake in a preheated oven at 400 degrees F (200 degrees C) for 15-18 minutes, or until golden brown and puffed.