Step 1In a large bowl, whisk together the water, oil, sugar and yeast. Set aside the mixture for 15 minutes to hydrate the yeast.
Step 2Stir in the salt and two-thirds of the flour, 1 spoonful at a time, to achieve a very soft dough.
Step 3Turn out the dough onto a well-floured board and continue kneading in additional flour until a soft, sticky dough is achieved. The dough should be very moist; the more flour added, the heavier the dough and final crust.
Step 4Place the dough in a large, greased bowl and cover loosely with plastic wrap. Set aside the dough until it doubles in volume, about 1 to 1 1/2 hours. (While the dough is rising, make the tomato and pesto sauces, and assemble the toppings.)
Step 5When the dough has doubled in size, punch it down and turn it out onto a floured board. Halve the dough and roll each half into a ball. Place on the floured board and lightly flour the tops. Loosely cover each ball with plastic wrap and set aside until almost doubled again.
Step 1In a medium saucepan heated over medium heat, add the oil, garlic and onion. Cook, stirring occasionally, until the onion is translucent, careful not to burn the garlic.
Step 2Stir in the tomatoes, tomato paste, salt and sugar. Cover the pan and gently simmer for 15 minutes, stirring occasionally. Remove from heat and stir in the herbs.
Step 3Pass the sauce through a food mill, or puree using a food processor or blender. This makes 1 scant cup tomato sauce. Cover and refrigerate until needed.
Step 1In a large mortar and pestle or using a food processor, grind together the garlic, pine nuts, chile flakes, Parmigiano and olive oil. Add the basil leaves, a few at a time, until they are incorporated and a coarse paste is formed.
Step 2Taste and adjust seasoning as desired with additional Parmigiano, lemon juice, pine nuts or olive oil. Thin as desired with a little olive oil.
Step 3Refrigerate the pesto in a nonreactive container with the surface covered with plastic wrap until needed; the pesto will keep for up to 1 day.
Step 1Place a pizza stone on a rack in the oven, and heat the oven to 450 degrees.
Step 2Roll out each dough ball to a 12- to 14-inch round and place on a floured pizza peel. Divide the sauce between the two rounds, ladling evenly over each round and leaving a 1-inch border around each.
Step 3Sprinkle the mozzarella and feta evenly over the pizzas, and garnish the top of each pie with the tomatoes, olives and artichokes.
Step 4Gently slide the first pizza on the stone. Bake until the crust is puffed and golden-brown, about 8 to 10 minutes (timing will vary depending on the thickness of the pie and heat of the oven). Remove the pizza and slide the second into the oven. Cool the pizzas slightly, then spoon the pesto over and slice. Serve immediately.