Lightly butter the bottom and sides of a 9×13-inch baking pan and set aside. In a small pot, melt 4 1/2 tablespoons of unsalted butter and add 2 tablespoons of minced garlic. Cook over low heat for 1-2 minutes until the garlic is fragrant. Remove from heat and let it cool while preparing the dough.

In a stand mixer bowl, combine 3 1/2 cups plus 1 tablespoon all-purpose flour, 3 tablespoons chopped parsley, 1 tablespoon sugar, 2 1/4 teaspoons instant yeast, and 1 teaspoon sea salt. Add the warmed milk (110°F), 1 large room temperature egg, and the cooled garlic butter mixture. Mix using the dough hook attachment on low speed.

Knead the mixture for about 10-12 minutes, or until the dough is smooth, soft, and slightly tacky. If the dough feels too sticky, gradually add flour a tablespoon at a time until manageable but still moist.

Divide the dough into 12 equal portions of approximately 67 grams each. Shape each portion into a smooth ball by pinching the seam at the bottom tightly. Arrange the dough balls in the prepared baking pan evenly spaced.

Cover the pan tightly with plastic wrap or a clean kitchen towel and place it in a warm area. Allow the rolls to rise until doubled in size, about 1 to 2 hours.

Preheat your oven to 350°F (180°C). Lightly brush the tops of the risen rolls with the beaten egg to help achieve a glossy, golden crust.

Bake in the preheated oven for 20-30 minutes until the rolls are golden brown on top and sound hollow when tapped.

While baking, melt 2 tablespoons unsalted butter in a small pot. Stir in 1 tablespoon minced garlic and cook for 1-2 minutes until fragrant. Remove from heat and mix in 1 tablespoon chopped fresh parsley.

Remove rolls from the oven and immediately brush them generously with the freshly prepared garlic butter. Sprinkle flaky sea salt over the tops as desired and serve warm for the best flavor and texture.