Place a cast-iron pizza pan or pizza stone in oven, and preheat oven to 550°F. (Do not remove pizza pan while oven preheats.) Heat anchovies and 1 cup of the oil in a small skillet over low; cook, stirring occasionally, until anchovies completely break down and look melted, about 10 minutes. Transfer to a bowl to cool completely, about 30 minutes.
Reserve 2 tablespoons of the anchovy oil. Place remaining 1 cup anchovy oil in a food processor; add garlic and 1/2 cup of the pistachios. Process until finely chopped, about 15 seconds. Add oregano and 1 cup each of the basil and parsley; process just until smooth, about 15 seconds. Transfer mixture to a medium bowl; stir in olives, 1/2 teaspoon of the salt, 1/2 cup of the feta, and 1 1/2 tablespoons of the lemon juice. Set aside.
Sprinkle a pizza peel with cornmeal; top with 1 pizza dough round. Brush dough with 1 tablespoon of the oil, and sprinkle with 1/4 teaspoon of the salt. Slide pizza onto preheated pizza pan.
Bake in preheated oven until golden brown, 8 to 10 minutes. Transfer crust to a cutting board. Repeat with cornmeal and remaining dough round, oil, and salt.
Spread pesto over crusts (about 1 cup each), leaving a 1-inch border. Toss together arugula, reserved 2 tablespoons anchovy oil, and remaining 1 tablespoon lemon juice and cup each basil and parsley. Top pizzas evenly with arugula mixture and remaining 1/2 cup each feta and pistachios.