Lightly butter the bottom and sides of a 9x13 pan and set aside.

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

Combine 3 1/2 cups of flour, 3 tablespoons of parsley, 1 tablespoon of sugar, 2 1/4 teaspoons of yeast, and 1 teaspoon of salt in a large mixing bowl.

Pour in the warmed milk, the egg, and the garlic butter you just made. Knead on low speed with the dough hook until smooth (about 10-12 minutes). Add more flour if dough is too sticky.

Cut dough into 12 equal portions (around 67 grams each) and shape into smooth balls.

Place each ball into the prepared baking pan, cover, and set aside to rise in a warm place until doubled in size (about 1-2 hours).

Preheat the oven to 350°F (180°C).

Brush the tops of the rolls with the beaten egg wash.

Bake for 20-30 minutes or until golden brown.

Melt the remaining 2 tablespoons of butter in a pot, add the minced garlic, and cook until fragrant (1-2 minutes). Stir in parsley.

Brush warm buns with garlic butter and sprinkle with flaky sea salt before serving.