Cut the top off the head of garlic. Place on a piece of foil and drizzle olive oil over garlic. Wrap up the foil around garlic and place in a preheated 375° F oven and bake for 45-60 minutes until garlic cloves are soft. Let cool for 10 minutes. Squeeze garlic cloves out of their skin and mash to a smooth paste. Set aside.
In a large bowl, combine 1-1/2 cups flour, sugar, undissolved yeast, salt, and herbs. Heat milk, water, and 2 tablespoons butter until very warm (120° F to 130° F); stir into dry ingredients. Stir in egg, roasted garlic and Parmesan cheese. Stir in enough remaining flour to make a soft dough. Knead on lightly floured surface until smooth and elastic, about 8-10 minutes. Cover; let rest on floured surface for 10 minutes.
Divide dough into 3 equal pieces. Roll each piece into a 30-inch rope. Braid ropes; pinch each end to seal. Tie a knot in the center of the braid; wrap ends around knot in opposite direction, and tuck under to make round loaf. Place on greased baking sheet. Cover; let rise in a warm draft-free place until doubled in size, about 45 minutes.
Preheat oven to 375° F. Bake for 35 minutes or to an internal temperature of 180° F, covering with foil during the last 15 minutes to prevent excessive browning. Melt remaining butter and brush over loaf. If desired, sprinkle with additional herbs and Parmesan cheese. Remove from baking sheet; let cool on wire rack.