First, prepare the red bell pepper: slice off the top, discard the seeds and core, then cut it into 1/4-inch rings. Brush these rings lightly with 1 tablespoon extra virgin olive oil.
For the pesto sauce, heat a dry frying pan and quickly toast the pine nuts until fragrant, which should take less than 3 minutes. Transfer them to a food processor.
Add the basil leaves, toasted pine nuts, 1/4 teaspoon salt, and 1 pinch pepper to the food processor. Pulse, slowly adding the remaining 4 tablespoons of extra virgin olive oil through the feed tube until you have a smooth puree. Stir in the mayonnaise and pulse 2-3 more times, scraping down the sides to ensure everything is well combined.
Now, cook the burgers. Lightly coat your grill with non-stick spray. Cook the patties over low to medium heat for about 8-10 minutes on each side. Try to avoid flipping them too often, as turkey burgers can be prone to crumbling. If you see mozzarella starting to ooze out, it's a good sign your burger is almost done.