Instructions

1. To make the spread, place the olives, tomatoes, and basil in the bowl of a food processor. Puree until combined and spreadable but with a little texture remaining. Set aside.
2. Preheat a grill pan over medium-high heat. Lightly grill each bun on the cut side until golden brown and toasted, about 1 minute. Set aside.
3. Divide the meat into 4 equal patties. Season the patties evenly with 1 teaspoon salt and the pepper. Grill over medium-high heat for 2 to 3 minutes per side for medium.
4. Spread the bottom buns with some of the olive-tomato mixture. Remove the burgers from the grill and place each patty on top of a bottom bun. Top the burgers with the shavings of Parmesan.
5. Meanwhile, preheat a 10-inch nonstick skillet over medium heat. Add the oil and crack the eggs into the pan, being careful not to break the yolks. Sprinkle with the remaining 1/2 teaspoon salt. Reduce the heat to medium-low and cook the eggs gently until the whites are set but the yolk is still runny, about 3 minutes.
6. Place a sunny-side-up egg on each patty, then add a few pieces of arugula. Add the tops of the buns to the burgers and serve.