In a small bowl, microwave the milk, water, and sugar for 45 seconds, until it is warm but not hot. Add in yeast and stir. Allow to sit for 5 minutes. The mixture should foam, signaling that the yeast is active. Then, stir in oil, egg, and salt.
Pour mixture into a standing mixer fitted with a dough hook. Turn the mixer to low speed and slowly add the flour 1 cup at a time. Let the dough knead for 5-10 minutes. Dough should be smooth and slightly sticky but firm enough to hold together in a ball.
Coat a large bowl with oil and place the dough into the bowl. Cover with plastic wrap and set in a warm area to proof for 1 hour or until the dough has doubled in size.
While dough is proofing, make your pesto. In a food processor, combine all ingredients except parmesan, pulsing until they come together. Set aside.
Line two bread pans with parchment paper. Set aside.
Flour a large work surface. Roll the dough out into a 12 inch by 20 inch rectangle. Smear the pesto over the dough, coating it evenly. Sprinkle the parmesan on top.
Roll the dough into a log (so the log is 20″ long). Using a serrated knife, cut the dough in half lengthwise (you will end up with two very long half cylinders). Carefully fold one half-cylinder over the other, keeping the cut side face up, the repeat with the other side, forming a long twist.
Cut the twist in half so you have two 10″ twists. Carefully place each twist into the lined bread pans, then loosely cover with plastic wrap or a kitchen towel. Allow dough to rise a second time in a warm place for 45 minutes.
Preheat the oven to 375°F. Bake for approximately 35 minutes, until the top is nicely golden brown. You’ll want a deep golden color, as the outside of the loaf can be cooked and the inside still doughy. Use your discretion.