In a small container, mix 20 g sourdough starter, 40 g all-purpose flour, and 40 g water. Cover and let rise until bubbly, active, and roughly doubled.
In a large mixing bowl, add the whole milk, granulated sugar, active sourdough starter, flour, egg, and salt. Mix until a rough dough forms.
Knead the dough until it begins to come together, then add the room-temperature butter.
Continue kneading until the butter is fully incorporated and the dough is soft, elastic, and slightly tacky.
Cover the dough and let it rise at room temperature until puffy and increased in size.
In a small bowl, mix the softened butter, chopped parsley, minced garlic, and salt until smooth.
Turn the dough out onto a lightly floured surface. Divide into equal pieces.
Flatten each dough piece, spread with garlic herb butter, and sprinkle with grated cheese.
Fold each piece and arrange the folded pieces upright in a greased loaf pan.
Cover the loaf and let it proof until puffy and nearly filling the pan.
Preheat the oven to 350°F (175°C).
Whisk 1 egg with 1 tablespoon water, then brush the egg wash over the top of the loaf.
Bake for 30 to 40 minutes, or until golden brown and cooked through.
Brush the warm bread with melted butter and sprinkle with flaky salt.
Let cool slightly, then pull apart and serve warm.