In the bowl of a stand mixer, mix together the warmed milk, sugar and yeast. Cover the bowl and set aside to activate the yeast. Yeast mixture is ready to be used when it appears foamy, about 8-10 minutes.
Into the bowl with the activated yeast mixture, add the egg, flour and salt.
With the dough hook attached to the stand mixer, mix the dough ingredients together at low speed just until a dough starts to form.
At this point, mix in the butter a little at a time. Once the butter is fully incorporated into the dough, increase mixer speed to medium and knead the dough for 5-6 minutes. The resulting dough should be smooth and slightly tacky to touch.
Using lightly greased hands, transfer the dough into a large greased mixing bowl, cover with plastic wrap and set aside in a warm area to rise until doubled in size, about 60-90 minutes.
As the dough rises, make the garlic butter. In a small mixing bowl, add the butter, minced garlic, fresh parsley, salt and black pepper. Mix well to combine and set aside.
Deflate risen dough, cover the bowl and let the dough rest for about 5 minutes. On a lightly floured surface, roll the dough into roughly a 12x16 inch rectangle. Spread about 2/3 of the garlic butter over the dough and evenly top with the mozzarella and parmesan cheese.
Starting on the long end of the dough, roll it into a log shape. Use a serrated knife to evenly cut into (8) 2 inch pieces. Transfer into a generously greased large (9x13 inch) baking pan (alternatively, you may line a large baking pan with parchment paper). Cover baking pan with plastic wrap and set aside to let the rolls rise until doubled in size, about 25-35 minutes.
As the rolls rises, preheat oven to 350F. In a small dish, whisk together the egg and water. Use a pastry brush to lightly apply the egg wash over the risen rolls. Bake rolls, rotating the pan halfway through bake time, until golden brown, about 25-30 minutes or until a toothpick inserted into one of the rolls comes out clean. (If rolls start to get too brown before it is fully baked through, tent the pan with aluminum foil to prevent over browning).
Melt remaining garlic butter and generously brush over the warm rolls before serving. Enjoy!!