Preheat the oven to 350°F.
In a large mortar, mash together the garlic and salt. Add the pine nuts, basil, lemon juice, and 1/3 cup olive oil and continue mashing into a paste. Set aside.
Whisk the eggs together in a medium bowl. Add the pesto and Parmigiano Reggiano, and season with salt and pepper. Stir until combined.
Heat the remaining 2 tablespoons olive oil in a medium cast-iron skillet over medium-high heat. Add the eggs to the pan and cook for 4 minutes. Arrange the tomatoes over the top. Place in the oven and bake for 20 to 25 minutes. Remove from the oven.
Garnish with fresh basil leaves and sprinkle with additional Parmigiano Reggiano, if desired. Slice into wedges and serve immediately.