Stir the yeast and sugar into the warm water. Let sit until foam appears, about 5 minutes.
Measure the flour into a medium bowl and whisk in the salt. Make a well in the middle. In small intervals, add the yeast to the flour, while stirring with your hand or a spoon to combine. Add more of the yeast, and continue to combine, while stirring to incorporate. Continue to stir until the dough starts to take shape and is sticky. Add additional flour and water as necessary by the spoonful and continue to mix.
With your hands, vigorously shake the dough around the bowl to tighten and form a ball shape. Add a few tablespoons of oil to a clean bowl and swirl around to coat the bowl. Add the dough and turn to coat with the oil. Cover the bowl with plastic wrap and let the dough rise in a proofing box or warm spot until doubled in size, about 1 hour 30 minutes.
Spread the dough onto a greased 13-by 18-inch baking sheet and gently punch down to expel any air, and then let it rise again for 1 hour to 1 hour 30 minutes.
Preheat the oven to 350 degrees F.
Ladle the sauce over the focaccia, and top with the vegetables, cheese and basil. Bake for 20 to 25 minutes.
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