1. Add all pesto ingredients to food processor. Pulse to combine. Process until smooth; set aside.
2. Sprinkle chicken with salt and pepper. Using three shallow bowls, add flour to one bowl. In next bowl, lightly beat eggs. In third bowl, combine breadcrumbs, 1/4 cup parmesan, oregano and basil. Coat each chicken breast in flour, shaking off excess. Dip chicken in egg and then breadcrumb mixture. Set coated chicken aside on a plate. 
3. Over medium heat, add vegetable oil to deep skillet. Heat oil to 350° F. Carefully, add chicken to pan, cooking two pieces at a time. Cook until golden on both sides, about 5 minutes per side. Transfer to foil-lined baking sheet. Repeat with remaining chicken. Place in 350° F oven for 10 minutes to finish cooking.
4. Remove chicken from oven. Top each chicken breast with 1/4 cup mozzarella. Divide remaining parmesan between chicken breasts. Heat broiler to high. Place chicken under broiler to melt cheese, about 2 minutes. Remove when cheese begins to bubble.
5. Combine arugula, olive oil and balsamic in bowl. Toss to combine. Spread a generous layer of pesto on buns. Add a chicken breast and some arugula to each sandwich. 

Calories: 1060, Fat: 65 g (14 g Saturated Fat), Cholesterol: 106 mg, Sodium: 1166 mg, Carbohydrates: 73 g, Fiber: 5 g, Protein: 51 g.