Combine the warm water, sugar, and yeast, in a small bowl. Set aside for 5-10 minutes until foamy.
Whisk together 3 ½ cups of the flour and the salt, in a large bowl. Once the yeast mixture has risen add it to the flour along with the olive oil. Mix with an electric mixer until the dough is smooth.
Flour a surface, and turn out the dough onto the surface, along with the remaining ½ cup flour. Knead for8 minutes. Add more flour a little at a time until the dough is smooth and elastic.
Oil a mixing bowl. Place the dough into the bowl, turning the ball once to oil both sides. Cover with a clean kitchen towel and set aside in a warm place to rise until doubled, 60-90 minutes.
Add vegetable oil to a heavy pot. Heat to 350°F over medium to medium-high heat.
Divide the dough into 6 equal pieces. Flatten each piece of dough by hand into a circle 8 inches across. Lower the dough into the hot oil. *Use extreme caution because the oil will splatter.
Fry the pizza dough for approximately 1 minute on each side until it begins to brown. With metal tongs remove thepizza crust from the oil and set on a cooling rack.
Repeat steps 6 and 7 with the remaining dough. Preheat the oven to 400°F degrees.
Top the pizza crusts with pizza sauce, slices of fresh mozzarella cheese, and any additional toppings you choose.
Place the pizzas on a cooking sheet and bake for 5-10 minutes.
Add the basil leaves just before serving.
Notes
Pizza will keep in the refrigerator for up to 3 days. It can also be frozen in an airtight container and will keep for 3 months in the freezer.