Place tomatoes on a double layer of paper towels; sprinkle with salt. Cover with additional paper towels. Meanwhile, in a small bowl, stir together pesto and mayonnaise. Preheat oven to 375°. On a floured sheet of parchment, roll dough into a 12-in. circle. Transfer to a baking sheet or pizza stone.
Spread pesto mixture over crust to within 2 in. of edge; top with ¼ cup feta. Arrange tomato slices on top; arrange cherry tomatoes on top, cut sides up. Sprinkle with remaining ¼ cup feta, Italian seasoning and pepper. Fold crust edges over filling, pleating as you go and leaving center uncovered. In a small bowl, whisk egg and water; brush edges of crust with egg wash and sprinkle with Parmesan.
Bake until crust is crust is golden brown, 35-40 minutes. Cool 10 minutes before slicing. If desired, top with basil before serving.
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