Preheat the oven to 400°F. Line a baking sheet with parchment paper and place the pizza crust on top.
In a small bowl, combine the cheeses and basil. Spread the cheese mixture evenly over the crust. Trim the ends of the tomato; reserve for another use. Cut the tomato crosswise into 4 even slices. Cut out the seeds and ribs, leaving 4 rings of tomato. Arrange 1 tomato ring in each quarter of the crust. Place the peppers around the tomatoes. Crack an egg into each tomato ring.
Bake until the eggs are set and cooked through, about 20 minutes. Sprinkle with salt, pepper and basil. Cut into slices and serve.