In a microwave safe dish, heat milk for 1 minute. Add 1/4 cup butter then let cool to 100°F.
Combine warm water, sugar and yeast; stir. Set aside for 5-10 minutes until foamy.
In large bowl, combine egg, milk mixture, yeast mixture, salt and half of the flour.
Continue to stir in enough of remaining flour until dough is no longer sticky and comes together in a ball. (Add extra flour, one tablespoon at a time, if needed.)
Transfer dough to a floured board. Knead 5 minutes or until dough is smooth and elastic. Add additional flour if needed. Transfer dough into large, oiled bowl. Drizzle oil on top of dough to coat. Cover bowl with a damp towel and place in a warm spot until dough doubles in size, approximately 90 minutes.
Transfer dough to flat surface and divide into 12 equal pieces. Dough will be slightly sticky and elastic.
Prepare a 9x13-inch glass baking dish: line with greased parchment paper.
Flatten each portion and fill with a dollop of pesto and 6-8 mozzarella pearls in center of dough. Fold edges over cheese/pesto into a ball shape and pinch to close. NOTE: Do not let oil from pesto drip onto the outer edges or the dough will not stick together. Transfer rolls to prepared baking dish. Cover with towel. Place dish in warm spot and allow rolls to double in size, about 60 minutes.
Preheat oven to 400ºF
Brush rolls with egg wash and sprinkle tops with Parmesan cheese (optional).
Bake for 12-14 minutes, until golden brown.
Remove from oven, brush with melted butter and sprinkle with sea salt. Let rolls rest for 5 minutes before serving.