Lightly butter the bottom and sides of a 9x13 inch pan, then set it aside.

In a small pot, melt 4 1/2 tablespoons of butter and add 2 tablespoons of minced garlic; cook for 1-2 minutes until fragrant, then remove from heat and cool.

In a stand mixing bowl, combine flour, 3 tablespoons parsley, sugar, yeast, and salt. Add the warmed milk, 1 egg, and the cooled garlic butter, then knead on low speed with a dough hook for 10-12 minutes until the dough is smooth and tacky.

Cut the kneaded dough into 12 equal portions, shaping each into a smooth ball and pinching the seam tight at the bottom. Place the shaped dough balls in the prepared pan, cover with plastic wrap or a kitchen towel, and let rise in a warm place for 1-2 hours until doubled in size.

Preheat the oven to 350°F (180°C).

Brush the tops of the risen rolls with the beaten egg wash, then bake for 20-30 minutes, or until golden brown.

While the rolls are baking, melt 2 tablespoons of butter in a pot, add 1 tablespoon of minced garlic, and cook for 1-2 minutes until fragrant; remove from heat and stir in 1 tablespoon of parsley. Immediately brush the warm, baked rolls with the garlic butter, sprinkle with flaky sea salt if desired, and serve warm.