Add sugar and yeast to the bowl of a stand mixer fitted with hook attachment. Pour in warm milk, then allow to sit, until foamy. Add bread flour, eggs, and kosher salt, then mix on low speed. Add softened unsalted butter to the dough, then increase the speed to medium high. Continue kneading.
Grease a large bowl with butter or oil, then transfer the dough inside. Drizzle with olive oil, then cover with a kitchen towel. Allow the dough to rise until it doubles in size.
Turn the dough out onto a lightly floured surface. Cut the dough in half, then into quarters. Fold the edges of the dough towards the center, then gently roll each into a smooth ball. Arrange the rolls in the prepared pan, cover again with a kitchen towel, then allow to rise.
Whisk egg and milk in a small bowl, then brush on top of the proofed rolls.
Transfer to the oven and bake until the rolls have puffed up and are golden brown. Immediately brush the rolls with melted butter, then sprinkle with grated Parmesan, chopped fresh parsley, and flaky sea salt. Enjoy warm or at room temperature!