In a large bowl, whisk together flour, sugar, yeast, and salt.
In a separate bowl, whisk together warm milk, melted butter, and egg.
Pour the wet ingredients into the dry ingredients and mix until a shaggy dough forms.
Turn the dough out onto a lightly floured surface and knead for 8-10 minutes, or until smooth and elastic.
Place the dough in a lightly greased bowl, cover with plastic wrap, and let rise in a warm place for 1-1.5 hours, or until doubled in size.
While the dough is rising, prepare the filling: In a small bowl, combine softened butter, minced garlic, shredded cheeses, and chopped parsley. Mix well.
Once the dough has risen, punch it down and roll it out into a rectangle, about 12x18 inches.
Spread the garlic cheese filling evenly over the rolled-out dough, leaving a small border on one of the long edges.
Starting from the long edge with the filling, tightly roll up the dough into a log.
Cut the log into 12 equal slices.
Arrange the rolls in a greased 9x13 inch baking dish, cut-side up. Cover and let rise for another 30 minutes.
Preheat oven to 375°F (190°C). Bake for 15-20 minutes, or until golden brown and heated through.
Let cool slightly before serving.
Notes
For an extra garlicky flavor, brush the tops of the rolls with melted garlic butter after baking.