Preheat and prepare:
Preheat the oven to 350°F (180°C). Lightly butter the bottom and sides of a 9×13-inch pan and set aside.
Cook the garlic butter:
In a small pot, melt the 4½ tablespoons of butter and add the minced garlic. Cook for 1-2 minutes until fragrant. Remove from heat and let it cool slightly.
Make the dough:
In the bowl of a stand mixer, combine the flour, chopped parsley, sugar, yeast, and sea salt.
Add the warmed milk, egg, and cooled garlic butter. Use the dough hook attachment and knead on low speed for 10-12 minutes until the dough is smooth and slightly tacky. If the dough feels too sticky, add flour 1 tablespoon at a time until you have a workable consistency.
Shape the dough:
Divide the dough into 12 equal portions (about 67 grams each for accuracy). Roll each portion into a smooth ball, pinching the bottom to seal.
Arrange the dough balls in the prepared baking pan, leaving some space between them. Cover with plastic wrap or a clean kitchen towel, and let rise in a warm spot for 1-2 hours until doubled in size.
Bake the rolls:
Once the dough has risen, brush the tops of the rolls with the beaten egg wash.
Bake for 20-30 minutes or until the tops are golden brown.
Make the garlic butter topping:
While the rolls are baking, melt the 2 tablespoons of butter in a small pot and add the minced garlic. Cook for 1-2 minutes until fragrant, then stir in the chopped parsley.
Finish and serve:
When the rolls come out of the oven, immediately brush them with the garlic butter mixture. Sprinkle with flaky sea salt, if desired, and serve warm.