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

In a small pot, melt 4.5 tablespoons butter and add 2 tablespoons minced garlic. Cook over medium heat for 1-2 minutes, stirring constantly, until the garlic is fragrant and softened but still pale golden (not brown, which tastes bitter). Remove from heat and let cool while you prepare other ingredients.

In a stand mixer bowl, combine flour, parsley, sugar, yeast, and salt. Add the warmed milk (110°F - no hotter than 120°F or it will kill the yeast), egg, and cooled garlic butter. Attach the dough hook and knead on low speed for 10-12 minutes until the dough is smooth, soft, and slightly tacky. If the dough sticks to the bowl, add flour one tablespoon at a time.

Divide the dough into 12 equal portions (67 grams each for accuracy). Shape each piece into a smooth ball by cupping your hand over it and rolling in circular motions on an unfloured surface. Pinch the bottom seam together and place seam-side down in the prepared pan. Cover with plastic wrap and let rise in a warm place (75-80°F) until doubled in size and puffy, about 1-2 hours.

Preheat the oven to 350°F (180°C). Brush the tops of the risen rolls with beaten egg wash.

Bake the rolls for 20-30 minutes until the tops are golden brown and sound hollow when you tap the top of a roll.

While the rolls bake, make the garlic butter by melting 2 tablespoons butter with 1 tablespoon minced garlic in a small pot. Cook for 1-2 minutes until fragrant, then remove from heat and stir in parsley.

When the rolls come out of the oven, immediately brush them generously with the garlic butter and sprinkle with flaky sea salt. Serve warm.