Wipe the mushrooms with a damp paper towel to clean and dry well. Do not rinse under water. Brush the tops of the mushrooms with canola oil and set aside.
In a mixing bowl, mix together the pesto sauce, Parmesan, sun-dried tomatoes and the bread crumbs. Season with salt and pepper, to taste, and mix thoroughly. Divide the mixture evenly into the 4 mushroom caps and top each with some mozzarella cheese. Put on the grill and cook until cheese is melted, about 10 minutes. Transfer to a serving dish and serve immediately.