Preheat gas grill to medium by turning on all burners and closing lid. Meanwhile, for pesto, toast the pine nuts in a small fireproof skillet, stirring until lightly toasted, about 5 minutes. Remove from heat; place nuts on small plate to cool. In a food processor, add cooled pine nuts, tomatoes, Parmesan cheese, 2 tablespoons oregano, 2 tablespoons parsley, lemon juice, 3 teaspoons garlic, 1/2 teaspoon salt, and 1/4 teaspoon ground black pepper; process until finely chopped, about 30 seconds. Slowly add 2/3 cup olive oil; process until completely combined. Scrape pesto into a small bowl; cover with plastic wrap.
For herbed oil, add the remaining 1/3 cup olive oil, 1 teaspoon garlic, 1 tablespoon basil, 1 tablespoon oregano, crushed red pepper, and 5 crushed peppercorns to a fireproof sauce pan. Heat and simmer for 10 minutes. Into a small bowl, carefully strain oil through a fine-mesh strainer to remove solids; reserve 2 1/2 tablespoons herbed oil for salad. Set aside remaining oil for buns.
To make the buns, lightly flour work surface. With floured fingers, press pizza dough in a 10-inch circle. For six buns, cut dough in half; cut each half in three wedges. Handling as little as possible, press each wedge into a 4-inch round. Let stand for 30 minutes at room temperature. Brush grill rack with vegetable oil. Place dough rounds on the rack; close lid for 1 to 2 minutes. Open lid. Using tongs or spatula, carefully turn buns; close lid and cook for 1 to 2 minutes longer. Brush with herbed oil and turn as necessary to achieve even browning and nice grill marks;, 4 to 6 minutes total. Remove buns from grill; place on rack to cool. Sprinkle buns with 1 1/2 teaspoons coarse salt, about 1/8 teaspoon salt per bun side.
Increase grill temperature to medium-high. For patties, place beef in a large chilled bowl. Add sausage, pancetta, shallot, 2 teaspoons garlic, 2 tablespoons parsley, 2 teaspoons salt, and 1 teaspoon ground black pepper. Using a large fork, gently mix to incorporate ingredients; do not overmix. Shape meat mixutre in six equal-size patties, about 4 1/2-inches in diameter. (Because the buns are rustic, burgers need not be perfectly round.) If necessary, clean grill rack and brush with vegetable oil. Grill patties on hottest part of grill, searing until browned, 1 to 2 minutes per side. Using a spatula, move patties to cooler part of grill; continue grilling until desired doneness, 4 to 6 minutes per side for medium. During the last few minutes of grilling, top each patty with thinly sliced mozzarella cheese and allow to melt. Transfer burgers to platter; loosely cover with foil and allow to rest. Slice buns; brush cut sides with herbed oil. Lightly toast cut sides on grill, about 2 minutes.
For greens, in a medium bowl whisk reserved 2-1/2 tablespoons herbed oil, vinegar, Dijon mustard, 2 tablespoons basil, 2 teaspoons oregano, 1/2 teaspoon salt, and 1/4 teaspoon ground black pepper. Toss in arugula.
To build burgers, evenly divide pesto among cut sides of buns, a generous 1 1/2 tablespoons each side. Arrange 1/4 cup greens on bun bottoms. Place burgers on greens, evenly layer remaining greens, then top with buns.
Have fun! Grilling is a hands-on sport and an opportunity to have some fun with your food and to entertain friends and family in the process! During a kitchen renovation, I went without the use of my kitchen for three months. It was then that my grill and I became acquainted and I experimented with the many things I could cook on it: Pizza, bacon and eggs, skillet cake ... and I have been a grilling fanatic ever since!