Preheat oven to 400-425°F. Coat a cooking pan with 1/3 cup extra virgin olive oil. Place chicken breasts on a flat surface and pound to a thickness of ¼ inch. Place two slices of genoa salami on each chicken breast half.
Mix parmesan cheese, cubed mozzarella, cubed fontina and shaved asiago in a bowl with a table spoon of olive oil, and then spread over the salami. Roll up the chicken starting with the shorter end. Place a toothpick or skewer in two or three places to hold chicken together.
Heat chicken in a pan with olive oil over medium heat until browned. Remove chicken and place on a baking sheet. Reserve brown bits in pan for gravy.
Put remaining olive oil in cooking pan with butter, flour, 1 tablespoon basil and 1 tablespoon parsley. Whisk together until combined. Remove gravy from pan and set aside.
Julienne the remaining 4 slices of salami, fry until crisp, remove and reserve fat in pan. Add mushrooms and garlic to pan, sauté then remove. Add to gravy mixture that was set aside.
Drizzle remaining basil and parsley over chicken on baking sheet. Bake for 15-20 minutes until chicken is cooked though. Spoon gravy on plate, slice chicken breasts into medallions and arrange on plate. Top with fried salami strips.