Apply a light coat of butter to the bottom and sides of a 9x13-inch pan, then set it aside.

In a small saucepan, heat 4 1/2 tablespoons of butter until melted, then incorporate 2 tablespoons of minced garlic. Sauté for 1-2 minutes until aromatic. Take off the heat and let it cool while you measure the remaining ingredients.

In the bowl of a stand mixer, whisk together 3 1/2 cups + 1 tablespoon of flour, 3 tablespoons of parsley, the sugar, yeast, and salt. Pour in the warm milk, the egg, and the cooled garlic-butter mixture. Using the dough hook on low speed, knead until the dough is smooth, soft, and slightly sticky, which should take about 10-12 minutes. If it's overly sticky, incorporate additional flour one tablespoon at a time until manageable.

Divide the kneaded dough into 12 even pieces (each around 67 grams). Roll each piece into a taut ball, pinching the bottom to seal. Arrange them in the prepared pan. Cover with plastic wrap or a clean towel and allow to rise in a warm spot until doubled in size, roughly 1-2 hours.

Set the oven to 350°F (175°C).

Lightly coat the roll tops with the beaten egg. Bake for 20-30 minutes, until they turn golden brown.

While the rolls are baking, prepare the garlic butter: in a small pot, melt 2 tablespoons of butter, add 1 tablespoon of minced garlic, and cook for 1-2 minutes until fragrant. Remove from heat and mix in 1 tablespoon of chopped parsley.

As soon as the rolls emerge from the oven, brush them immediately with the garlic butter. If desired, sprinkle with flaky sea salt and serve warm.