Preheat the oven to 350°F (180°C). Lightly grease a 9x13 pan and set it aside.
Melt the butter in a small pot and add the minced garlic. Cook for 1-2 minutes, until fragrant. Remove from heat and cool.
In a stand mixer bowl, combine flour, parsley, sugar, yeast, and salt. Add the warmed milk, egg, and garlic butter. Knead with a dough hook on low speed until the dough is smooth, soft, and tacky (10-12 minutes). Add flour 1 tbsp at a time if dough is too sticky.
Divide dough into 12 equal portions (approximately 67 grams each). Shape each into a smooth ball, pinching the bottom to seal. Place rolls in the prepared pan.
Cover with plastic wrap or a kitchen towel and let rise in a warm place until doubled in size (1-2 hours).
Brush the tops of the rolls with beaten egg. Bake for 20-30 minutes, or until golden brown.
While baking: Prepare garlic butter by melting butter in a pot, adding minced garlic, and cooking for 1-2 minutes. Stir in parsley and set aside.
After baking, brush rolls with garlic butter and sprinkle with flaky sea salt if desired. Serve warm!