Combine the warm water, granulated sugar, honey, and yeast together in a large bowl. If using active dry yeast, let it proof for 5 minutes until foamy. If using instant yeast, proceed immediately to the next step.
Add the olive oil, salt, and flour, and mix until combined either using a large wooden spoon or a stand mixer fitted with a dough hook attachment.
Knead for 5 minutes using the dough hook attachment on medium-low speed or by hand until the dough is smooth and elastic. If it is too sticky, you can add another ¼ cup of flour.
Add a little olive oil to a large, clean bowl and transfer the ball of pizza dough to that bowl, turning to coat in the oil. Cover with plastic wrap and let rise for about 60-90 minutes until doubled in size.
While the dough is rising, preheat the oven to 500 degrees F (or as high as your oven will go). If you have a pizza stone, place that on the bottom rack of your oven before heating the oven so it can get really hot at the same time.
When the dough has risen, divide it into two evenly-sized balls.
Working with one ball of dough at a time, dust the dough with a little flour, then flatten it into a round disc. Use your hands to press and stretch the dough into a large, thin circle. If the dough keeps shrinking back, let it rest for 5 minutes before trying again. I like to do this on a sheet of parchment paper to make it easy to transfer my pizza to the oven.
Spread ¼-1/2 cup of pesto sauce on each circle of dough, depending on how much you like. Then sprinkle the pizzas with a little cheese and arrange the sliced veggies on top. Sprinkle with the remaining cheese, then bake for 12-15 minutes until brown around the edges and cooked through.
Slice and serve!