Lightly butter the bottom and sides of a 9x13 baking pan and set aside.
In a small pot, melt 4 1/2 tablespoons butter and add 2 tablespoons minced garlic. Cook 1-2 minutes until fragrant, then remove from heat.
In a stand mixer bowl, combine the flour, parsley, sugar, yeast, and salt.
Add warmed milk, egg, and garlic butter mixture to the dry ingredients.
Knead on low speed with dough hook until smooth and tacky, about 10-12 minutes. Add flour tablespoon by tablespoon if dough is too sticky.
Divide dough into 12 equal portions (about 67 grams each), shape into smooth balls, pinch bottom seams tight.
Place dough balls into prepared pan, cover with plastic wrap or kitchen towel.
Let rise in a warm place until doubled in size, about 1-2 hours.
Preheat oven to 350°F (180°C).
Brush tops of rolls with beaten egg.
Bake rolls for 20-30 minutes until golden brown.
While baking, melt 2 tablespoons butter in pot, add 1 tablespoon minced garlic, cook 1-2 minutes until fragrant.
Remove from heat, stir in 1 tablespoon chopped parsley.
When rolls are out of oven, immediately brush with garlic butter.
Sprinkle with flaky sea salt if desired and serve warm.
