In a small bowl, combine the warm water, sugar, and yeast. Set aside for 5-10 minutes until foamy. Note: If the mixture doesn't become foamy, the yeast has probably expired; the dough won't rise without this step, so you may need to get a new packet of yeast.
In a large mixing bowl, whisk together 3 1/2 cups of the flour and the salt. Add the yeast mixture and olive oil. Mix with an electric mixer until the dough is smooth.
Turn the dough out onto a surface floured with some of the remaining 1/2 cup flour. Knead for 8 minutes, adding more flour a little at a time, until the dough is smooth and elastic.
Place the dough into an oiled mixing bowl, turning to oil both sides. Cover with a clean kitchen towel and set aside in a warm place to rise until doubled, 60-90 minutes. Note: I keep my house pretty chilly year-round, so in a pinch I will start my dryer on high, let it run for 5 minutes or so, then turn it off so it's not moving, and set my covered bowl upright in the well of the dryer. Make sure no one comes by to do laundry while you wait, though!
Divide the dough into 6 equal pieces. Flatten one piece of dough by hand into a circle 8 inches across.
Heat vegetable oil to 350°F in a heavy pot or cast iron pan with high sides on the stove over medium to medium-high heat. Very carefully lower the flattened pizza dough into the hot oil. Use extreme caution when frying—the oil tends to spit, so stand back and keep children away.
Fry the pizza dough on each side until it begins to brown, about 1 minute per side. Remove the crust from the oil with metal tongs and set on a cooling rack, or a plate with a paper towel to drain.
Repeat the flattening and frying of the remaining dough. If desired, add a garlic seasoning blend or grated parmesan to the freshly fried crusts.
Heat the oven to 400°F.
Top the fried crusts with pizza sauce, slices of fresh mozzarella, and any additional toppings you choose.
Bake the pizzas on a cookie sheet in a 400°F oven for 5-10 minutes, depending on your desired level of doneness.
Tear the basil leaves into bite-size pieces, then scatter over the pizzas before serving.