In the bowl of a stand mixer fitted with the paddle attachment, combine the warm milk, sugar, and yeast. Let stand until mixture is foamy, about 10 minutes.
With mixer on low speed, beat in 1 cup of the flour just until combined. Add the melted butter and eggs and beat until combined. Gradually add salt and remaining 2 to 2 ¼ cups flour, beating until a soft dough forms. (Dough will be sticky.)
Put the dough into a large greased bowl, turning to coat. Spray a large bowl with cooking spray. Loosely cover and let rise in a warm, draft-free place until doubled in size, about 1 hour.
Meanwhile, get the filling ready. In a small bowl, mix the butter, garlic, parsley and salt. In another small bowl, mix together the mozzarella and Parmesan cheese.
Punch down the dough, turn it out onto a flour-dusted surface and let rest for 10 minutes.
Roll dough into an 18x16-inch rectangle. Spread a thin layer of the butter mixture over the dough and sprinkle with cheese mixture. Using a knife or pizza wheel, slice dough lengthwise into 4 equal strips. Stack strips one on top of the other, and cut into 4x3-inch rectangles.
Spray an 9x5-inch loaf pan with cooking spray, and prop pan at an angle on a book or other object so pan is not flat. Stack rectangles on their sides in prepared pan. Loosely cover and let rise in a warm, draft-free place until doubled in size, 30 minutes to 1 hour.
Preheat oven to 350°F.
Drizzle melted. butter on top of loaf, and bake until golden brown, 30 to 35 minutes, covering with foil halfway through baking to prevent excess browning, if necessary. Let cool for 20 minutes before inverting onto a serving plate. Turn right side up to serve.