Prepare a medium-hot fire in a charcoal grill with a cover, or preheat a gas grill to medium-high.
Combine the mayonnaise and sun-dried tomatoes in a small bowl and mix well. Cover and refrigerate until ready to use.
Place the oil in a 12-inch fire-proof skillet and set on the grill rack. When the oil is hot, add the onion, bell pepper, and mushrooms. Cook, stirring frequently, until tender, about 5 minutes. Add the spinach leaves and cook until the spinach is wilted, about 3 minutes. Transfer to a large bowl and add the sirloin, feta cheese, and salt. With your hands or a large spoon, gently mix well, handling the meat as little as possible to prevent compacting it. Form into 6 equal-sized patties.
Pour the oil into the same skillet and place on the grill rack. When the oil is hot, add the prosciutto. Cook until well crisped, about 5 to 8 minutes. Transfer to a paper towel-lined plate to drain; set aside.
When the grill is ready, brush the grill rack with vegetable oil. Place the patties on the rack, cover, and cook until the bottoms are brown, about 4 minutes. With a spatula, turn over the patties and cook an additional 4 minutes, or until done to your preference (internal temperature of 160 degrees F). During the last few minutes of grilling, top each patty with a slice of mozzarella cheese until melted. Lightly brush the cut sides of the rolls with the 4 tablespoons olive oil. Rub the cut sides of the garlic over the insides of the rolls, then sprinkle and press with the Parmesan. Place the rolls, cut sides down, on the outer edges of the grill rack to toast lightly.
Spread the mayonnaise mixture on the insides of the warm toasted rolls. Layer the bottom rolls with divided greens, the patties, and divided frizzled prosciutto. Top with the remaining rolls, serve, and enjoy!