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

In a small pot, melt 4 1/2 tablespoons unsalted butter over low heat, then add 2 tablespoons minced garlic. Cook for 1–2 minutes until the garlic is fragrant.

In a stand mixing bowl, combine 3 1/2 cups + 1 tablespoon all-purpose flour, 3 tablespoons chopped parsley, 1 tablespoon granulated sugar, 2 1/4 teaspoons instant yeast, and 1 teaspoon fine sea salt. Whisk briefly to distribute evenly.

Add 1 cup warmed milk, 1 large egg, and the cooled garlic butter to the dry mix. Knead on low speed until the dough looks smooth and feels soft and tacky, about 10–12 minutes.

Cut the kneaded dough into 12 equal portions. Shape each piece into a smooth ball and place in the prepared pan.

Cover the pan with plastic wrap and set it in a warm place to rise until doubled in size, about 1–2 hours.

Preheat the oven to 350°F. Brush the tops with the beaten egg and bake for 20–30 minutes, until golden brown.

While the rolls bake, melt 2 tablespoons unsalted butter in a small pot, add 1 tablespoon minced garlic, and cook for 1–2 minutes until fragrant.

Immediately brush the rolls with the garlic butter when they come out of the oven. Sprinkle with flaky sea salt if desired.