Prepare the dough by combining together the water, yeast, olive oil and honey in a large mixing bowl, or in the bowl of a stand mixer fitted with the dough hook. Let yeast proof for 3-4 minutes, or until just foamy.

Add in the flour (starting with 2 1/2 cups) and salt, and mix until well-combined. Knead (either by hand, or using the mixer) for 5-7 minutes, until smooth and elastic. You may need to add more flour. Transfer the dough to an oiled bowl, cover, and let rise in a warm place for an hour.

In a large bowl, toss together the cherry tomatoes, red onion, mushrooms, peppers, olives, and squash. Add in the olive oil, salt and pepper, and toss to coat.

Spread the veggies on the prepared baking sheet. And roast in preheated oven for 12-15 minutes, or until all veggies are tender and beginning to brown. Remove from oven and set aside.

With five minutes left in the roasting time, preheat the grill to medium-high.

Divide the dough in half, and shape each into a ball. Place those dough balls on a platter along with the roasted veggies. In a small bowl, combine the chives, oregano, and parsely. Set on platter. Add the olive oil, pesto, and cheese to the platter.

Take the platter out to the grill. Working with one flatbread at a time, take the dough, and stretch it until it is about 12" around (no need to be precise about it). Brush one side with olive oil, and place that side down on the grill. Close the lid, and let cook for 1-2 minutes, or until the edges are just beginning to solidify.

Brush the top with olive oil, and flip the flatbread. Spread half the pesto on top, followed by half the veggies, and half the cheese. Close the lid, and let cook for an additional 2-3 minutes, or until the cheese is melted. Remove flatbread from the grill, sprinkle with half the herbs, and set aside. Repeat with second flatbread.