Rinse spinach; drain briefly, then place in large glass bowl. Microwave spinach, uncovered, on high just until wilted, 1 to 1 1/2 minutes. Drain, then squeeze out as much liquid as possible. Combine garlic, lemon peel, and crushed red pepper in processor; blend until garlic is finely chopped. Add spinach, 2 cups (packed) arugula, pine nuts, and lemon juice; process until coarse puree forms. With machine running, gradually add oil in thin stream and blend until almost smooth. Mix in cheese. Transfer pesto to small bowl; season with salt. Do ahead: Can be made 6 hours ahead. Cover; chill. Combine ground beef, 1 teaspoon salt, 1/2 teaspoon black pepper, and 6 tablespoons spinach-arugula pesto in large bowl; mix lightly with fingertips or fork just until incorporated. Form meat mixture into six 3/4-inch-thick patties. Place patties on platter. Do ahead: Beef patties can be made 6 hours ahead. Cover and refrigerate. Prepare barbecue (medium-high heat). Grill burgers to desired doneness, about 4 minutes per side for medium-rare. Grill buns, cut side down, just until lightly toasted, about 1 minute. Build burgers with pesto, patties, mozzarella cheese, tomatoes, and arugula. Cover with bun tops.