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

In a small pot, melt 4 1/2 tablespoons of the unsalted butter and add the minced garlic. Cook for 1-2 minutes or until the garlic is fragrant. Remove from heat and cool while you measure the other ingredients.

In a stand mixing bowl, combine 3 1/2 cups + 1 tablespoon of flour, parsley, sugar, yeast, and salt. Add the warmed milk, egg, and the garlic butter mixture from step 2.

Knead on low speed with the dough hook until the dough looks smooth and feels soft and tacky, about 10-12 minutes. If the dough is too sticky, add one tablespoon of flour at a time until you have a workable dough.

Cut the kneaded dough into 12 equal portions. Shape each piece of dough into a smooth ball, pinching the bottom where the seam comes together. Place the shaped dough balls into the prepared baking pan.

Cover with plastic wrap or a clean kitchen towel and set in a warm place to rise 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 the dinner rolls for 20-30 minutes, or until the tops are golden brown.

While the rolls are baking, make the garlic butter. Melt 2 tablespoons of unsalted butter in a pot and add the minced garlic. Cook for 1-2 minutes until the garlic is fragrant. Remove from heat and stir in the parsley.

When the rolls come out of the oven, immediately brush them with the prepared garlic butter. Sprinkle with flaky sea salt if desired, and serve warm!