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1. In a large bowl, combine 1-3/4 cups of the flour and the yeast; set aside. In a medium saucepan, combine milk, the water, butter, sugar, and salt; heat and stir just until warm (120 degrees F to 130 degrees F) and butter almost melts. Add milk mixture and egg to flour mixture. Beat with an electric mixer on low to medium speed for 30 seconds, scraping the bowl constantly. Beat on high speed for 3 minutes. Stir in pesto. Using a wooden spoon, stir in as much of the remaining flour as you can.
2. Turn out onto a lightly floured surface. Knead in enough of the remaining flour to make a moderately stiff dough that is smooth and elastic (6 to 8 minutes total). Shape dough into a ball. Place in a lightly greased large bowl, turning once to grease surface of dough. Cover bowl with plastic wrap; chill for at least 4 hours or up to 24 hours.
3. Punch down dough. Turn out onto a lightly floured surface. Divide dough in half. Cover; let rest for 10 minutes. Meanwhile, lightly grease eighteen 2-1/2-inch muffin cups.
4. Divide each dough half into nine portions. Divide each portion into three pieces. Shape each piece into a ball, pulling edges under to make a smooth top. Place three balls in each prepared muffin cup, smooth sides up. Sprinkle tops with Parmesan cheese. Cover and let rise in a warm place until nearly double in size (about 30 minutes).
5. Preheat oven to 375 degrees F. Bake for 12 to 14 minutes or until rolls are golden brown. Immediately remove rolls from muffin cups. Serve warm or cool on wire racks.