Preheat the oven to 450 degrees F. Place a 12-inch cast iron skillet on a rack in the upper half of the oven.
In a medium mixing bowl, whisk the flour and 1 teaspoon salt together until incorporated. Slowly pour in 1 cup of lukewarm water (you can test it on your forearm - it should feel slightly warm on your skin) while whisking. Stir until the liquid is absorbed and the batter is smooth. Allow to sit, covered, until the oven is hot. Note: the batter can also sit for up to 24 hours.
Stir in 2 tablespoons of olive oil into the batter. Remove the hot pan from the oven and add 2 tablespoons of olive oil - enough to coat the bottom with a thin layer. Swirl the oil around the pan so it’s well coated. Pour in the batter and swirl the pan so it forms an even layer.
Bake the chickpea crust in the oven for 12 minutes, until firmly set. Remove the pan from the oven and drizzle with the tomato sauce, spreading it evenly over the top leaving a 1/4 inch of room around the crust. Top with the tomato slices and dollops (or slices) of nut cheese.
Return to the top half of the oven and bake until golden brown on top and the tomatoes are tender, about 5-10 minutes.
Meanwhile, in a medium mixing bowl, toss the radicchio, arugula and lemon juice or vinegar with your hands until well-combined. Season generously with salt.
Remove the pizza from the oven and top with the salad mixture, basil leaves and garnish with red pepper flakes. Enjoy immediately!