Combine the warm milk, sugar, and yeast in a large bowl. Stir and let sit for 5 minutes until slightly foamy. Add the melted butter, egg, and salt. Gradually stir in the flour until a soft dough forms.
Turn the dough out onto a lightly floured surface and knead for about 8 to 10 minutes until smooth and elastic. Place in a greased bowl, cover, and let rise in a warm spot for 1 to 1½ hours until doubled in size.
Once risen, punch down the dough and transfer to a floured surface. Roll into a rectangle about 8×14 inches. Sprinkle the shredded cheese over the surface, roll tightly from the short end, and seal the seam. Place seam-side down in a greased loaf pan and let rise for another 30 to 45 minutes.
Preheat your oven to 350°F (175°C). Bake the loaf for 30 to 35 minutes until golden brown and hollow-sounding when tapped. Brush the top with melted butter after baking and let cool for at least 15 minutes before slicing.