Mix all the ingredients for the sauce in a bowl, cover, and set aside.
Turn on the grill to preheat to medium-high temperature. Grease the wire rack with oil.
Grease the mushroom caps with vegetable oil on all sides. Place the mushroom caps on the grill rack, fry for 5 minutes, then turn the caps upside down on the other side, pour Parmesan into the caps, cover with mozzarella slices, cook for another 5 minutes.
Grease each half of the buns with sauce. On each base rug, put one mushroom cap with cheese up, cover with a half-top.