In your breadmaker pan or stand mixer bowl, add the milk and the egg.
Add the flour to cover the entire surface of the liquid, sprinkle the sugar and salt around the edges, make a well in the center of the flour and add the yeast in the well.
Transfer the pan to your bread machine, and start the Dough program. If using a stand mixer, start the mixer at the lowest speed, and mix until a shaggy dough forms. Increase the speed one step up, and continue kneading.
After kneading for about 10 minutes, add the butter. let the Dough knead cycle finish. If using a mixer, continue kneading, until the butter is incorporated and the dough is smooth and elastic, another 5 to 8 minutes.
Transfer to an oiled bowl. You can leave it in the machine if you like, but I prefer to use a clear glass bowl like this so I can visually see the dough double in size.
When the Dough knead cycle finishes, transfer to an oiled bowl. Cover and let rest in a warm place, until doubled in size, 40 minutes to 1 hour.
While the dough rests, make the filling. In a small bowl, add the butter, garlic and parsley, and stir to thoroughly combine. Set aside until ready to use.
Gently punch down the dough to let out the air. Transfer the dough to a lightly dusted surface.
Roll out the dough into a 9-inch wide rectangle.
Make many thin cuts along the length of the dough.
Spread the garlic butter filling evenly over the bottom half of the dough. Top with the shredded cheese.
Starting from the bottom short edge, roll upwards into a tight log. Pinch the edges to seal.
Transfer to an 9- by 5-inch loaf pan. Cover and let rest, until puffy, about 20 minutes.
Preheat the oven at 180 degrees C (fan 160 degrees C).
While the loaf rests, make the glaze. In a small bowl, add the egg and milk, and beat until thoroughly combined.
Brush the glaze evenly over the loaf.
Bake in the oven, until golden brown, 25 minutes.
Brush with the melted butter. Remove from the pan and let cool on a wire rack.