To make the dough, dissolve the yeast and salt in a large bowl with the water. Add the flour and use some long chopsticks to stir together (okay they probably didn't use chopsticks in Italy during the 19th century, but they work so well because they don't have much surface area for the dough to stick to). Cover with plastic wrap and let it sit for at least 12 hours, or overnight in the fridge.
Cut a cross hatch on the bottom of the tomatoes (opposite of where the stem connected) to keep them from popping. Put some water in a pot and place a steamer rack in the pot. Place the tomatoes cut-side-up on the rack. Bring the water to a boil over high heat, then cover and steam the tomatoes until they are soft to the touch (about 10 minutes). Remove the lid and let them cool off enough to handle.
Remove the skins, then place them in a stainer and use a pestle or the bottom of the spoon to press the tomato through. It would be faster to use a blender first before straining, but Queen Margherita's chef in Naples didn't have a blender. Plus using a blender would break up the seeds which would add a slightly bitter taste to the sauce.
When you're ready to start making the pizzas, move your oven rack to the top position and turn the boiler on high. Put a cast iron pan over medium-high heat. Flour your hands and flat surface to roll the dough out on and cut off a ball of dough about the size of a large orange. Shape it into a ball, then roll it in flour to keep it from sticking. Use your hands or a rolling pin to flatten the ball into a round flat circle about the same diameter of your cast iron pan. Since it will be difficult getting the dough into the pan once the toppings are on, I recommend having your toppings at the ready, and putting the dough straight into the pan at this point.
Once the dough is in the pan, drizzle some olive oil on top, use a spoon to spread a few spoonfuls of tomato puree on top of the pizza. You want enough to make a very thin layer, but don't add too much or your pizza will be watery. Lay a few slices of mozzarella on top, then sprinkle with salt.
By the time you have all the toppings on the pizza the dough should have puffed up slightly and a peak under the pizza should reveal a crust that's just starting to brown. Transfer the pan to the broiler (directly under the heat source), and broil for one to two minutes, or until the edges of the crust are just starting to char and the center of the pizza is bubbly and caramelized. You may need to turn the pan part of the way through cooking to get even browing on the crust.
When it's done, use a spatula to transfer the pizza to a plate and sprinkle with basil. Personally I love lots of basil, so I like covering it with a mound of basil, but we're being traditional here, remember? There should be enough dough to make four pizzas and you can turn the leftover tomato puree into a pasta sauce with any leftover basil.