For the dough, combine the yeast and warm water in a large bowl. Give the yeast a stir to help dissolve it and let it do its thing for 5 minutes. You're giving it a little bit of a kick-start, giving it some room to activate, to breathe.
When the yeast has dissolved, stir in 3 cups of the flour, mixing gently until smooth. You're letting the flour marry the yeast. Slowly add 2 cups more flour, working it in gently. You should be able to smell the yeast working—that happy yeasty smell. Add the salt.
Turn the dough out onto a floured work surface and get to work. With the dough on the counter, start stretching it and folding it back on itself. Repeat the process until the dough is noticeably easier to handle, 10 to 15 times, then knead until it's smooth and stretchy, soft and still a little tacky. This should take about 10 minutes, but here, feel is everything.
Shape the dough into a ball and put it in a lightly greased big bowl. Roll the dough around to coat it with oil, then cover the bowl with plastic wrap and let the dough rest in a warm place until it doubles in size, 2 to 2 ½ hours. When you press the fully proofed dough with your finger, the indentation should remain.
Turn the proofed dough out onto a floured work surface and cut it into 4 pieces. Roll the pieces into balls and dust them with flour. Cover with plastic wrap and let them rest for another hour, or until they have doubled in size.
The dough is ready to be shaped, topped and baked.
Put a pizza stone in the oven and preheat the oven to 500°F. Let the stone heat for at least 30 minutes.
For the pizza, on a lightly floured surface, roll or stretch the dough into a 12-inch round, then transfer it to a floured pizza peel. Spread the Parmigiano evenly over the dough, leaving a ¾-inch border all around. Evenly distribute the red onion and rosemary over the cheese.
Slide the pizza onto the pizza stone and bake the pizza halfway, about 4 minutes. Sprinkle with the pistachios and continue cooking until the crust is golden brown, about 2 minutes more. Drizzle with extra-virgin olive oil and serve.
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