Place the water in a medium-large bowl, sprinkle in the yeast and sugar, and stir to dissolve. Let it stand 5 minutes, or until it begins to bubble.
Stir in the salt, oil, and 1 cup flour. Beat for several minutes with a wooden spoon.
Add the remaining flour 1/2 cup at a time, mixing after each addition. The dough will be soft, but should not be sticky.
Turn out onto a floured surface, and knead for several minutes. Place in an oiled bowl, cover with plastic wrap, and let it rise until doubled in bulk. This will take about 1 hour.
Punch down the dough, and return it to the floured surface. (This is the point at which you can freeze the dough for future use.) Divide it into four equal parts, knead each quarter for a few minutes, then let the four balls of dough rest for about 10 minutes. (This allows the gluten to relax, so the dough will stretch easily into shape.)
Patiently stretch each ball into a 6-inch circle. Sprinkle two baking trays with cornmeal, and place two circles on each. (Meanwhile, preheat the oven to 500 degrees F.) Top each pizza with any combination of toppings. Make sure the walnuts are on top, so they can toast.
Bake in the lower half of the oven for 10 to 12 minutes, or until the edges are crispy and brown. (If you're not sure whether or not it's baked through, you can take one pizza out of the oven and cut it in half. If it is still a little doughy on the inside, return it to the baking pan and bake a few minutes longer.) Serve hot, warm, or at room temperature.