Make the garlic butter. Melt the butter in a small saucepan over medium low heat, then add the garlic. Cook for about a minute or until the garlic is fragrant. Remove from heat, and pour ⅓ cup of the mixture into a separate measuring cup. You can cover the remaining butter in the pot and just let it hang out until later. Cool the 1/3 cup butter until warm, but not hot (somewhere between 105 and 115 degrees).
While the butter cools, add 3 3/4 cups of flour, yeast, sugar, and salt to the bowl of a stand mixer and mix on the lowest setting until combined. Add the warm milk, ⅓ cup of the butter garlic mixture, and egg and beat on low speed until flour is just combined.
Switch to the dough hook attachment and increase the speed to medium. Beat for 2 minutes. Scrape the sides of the bowl and the hook, then beat again on medium speed for about a minute. At this point, the dough should begin pulling away from the sides of the bowl. If the dough is still quite sticky, add flour, 1-2 tablespoons at a time, until it is tacky but no longer sticks to your fingers, beating well after each addition.
Transfer the dough to a lightly greased bowl and loosely cover with saran wrap or a towel. Place in a warm-ish spot to rise for 30 minutes.
Lightly punch the dough to release any air bubbles. Grease a 9 x 13 inch baking dish (or 2 8 inch square or 9 inch pie plates) with non-stick spray. Pinch pieces of dough off and form them into balls, about 2 inches around. If you have a kitchen scale, you can measure the dough balls to make sure they’re similar in size. Mine were about 2 ounces each. Tip: I like to fold/tuck the edges of the dough under itself to create a smooth top, then roll it between my palms for a few seconds to form the ball. Line the dough balls up in the prepared pan.
Cover the pan with a towel or saran wrap and place back in a warm spot to rise for another 30 minutes.
Preheat the oven to 375 degrees. Whisk together the egg and Tablespoon of milk, then brush it on the top of the rolls. Bake for 12-14 minutes, or until the rolls are glossy and golden brown on top.
If needed, re-melt the butter in the sauce pan while the rolls are baking. Remove the rolls from the oven and brush/pour the garlic butter over top. Sprinkle with grated Parmesan and parsley, if desired. Serve immediately or cool and store.