For the pizza dough: Place the warm water in a small bowl and stir in the yeast and sugar until dissolved. Cover the bowl with a clean towel and set aside until the liquid is foamy (about 10 minutes).
In a large bowl, combine 3 cups flour and 1 teaspoon salt. Make a well in the center of the flour and add the yeast mixture and oil. With a wooden spoon, beat the ingredients together to form a soft, sticky mass.
Turn the dough onto a well-floured work surface. With floured hands, knead the dough, adding more flour, a little at a time, until smooth and elastic, 8 to 12 minutes. Form the dough into a ball.
Place the dough in a large bowl that has been brushed with oil. Turn the dough to coat all sides with oil. Cover with wax paper and a clean towel, and place in a warm draft free place. Let the dough rise until doubled in size, about 1 hour.
With a floured fist, punch down the dough. Cover again and allow the dough to rise another 40 minutes.
The dough is now ready to be rolled out for a pizza.
For the pesto pizza: Place the rack in the center of the oven. Preheat the oven to 425 degrees F. Place the rolled-out pizza dough onto a 16-inch oiled pizza pan or screen.
Place the pesto onto the pizza dough and spread it around the surface of the pizza. Leave about a 1/2-inch border on the outside of dough.
Sprinkle the cheese evenly over the pizza. Place the chopped chicken on top of the cheese.
Bake 25 to 30 minutes. Check it occasionally to ensure it doesn't burn.
Remove the cooked pizza from the oven and let stand for 5 minutes (if you can wait that long). Sprinkle the basil on top of pizza, then cut and enjoy!