In the bowl of a stand mixer, combine the warm water, yeast, and sugar or agave. Let bloom for about 5 minutes. It should be very bubbly.
With the dough hook attached, add 1 teaspoon of salt, then begin adding the flour, about 1/2 a cup at a time. Letting the dough hook mix. Scrape down the sides of the bowl as needed.
Once all of the flour has been incorporated, let the dough hook knead the dough until it forms a dough ball. Then let it knead for about 2 minutes.
Remove the dough from the bowl, and then drizzle 2 tablespoons of the olive oil into the bowl. Put the dough back into the bowl, and flip to coat in the oil. Cover the bowl with a cloth.
Let the dough rise in a warm, dry place for about 1- 1 1/2 hours or until the dough has doubled in size.
Pour 2 more tablespoons into a 9x13-inch baking dish and spread it around. Turn the dough out into the baking dish.
Push the dough out flat into the baking dish, pressing it evenly into the entire dish. It will shrink up a bit, then press it back out again. Cover with a cloth and let rise again for about 45 minutes.
Toward the end of the second rise, preheat the oven to 425 degrees(F).
Once the dough has risen again, pour the remaining 2 tablespoons of olive oil over the top of the dough. Then create divots in the top using your fingers.
Press the tomatoes, olives, garlic, and capers into the top of the dough. Sprinkle the top with the remaining 1/2 teaspoon of salt, the red pepper, and the thyme.
Bake for 25-35 minutes or until the top is golden brown.