In a medium bowl, stir together the water, yeast, and sugar. Set aside for 5 minutes, until the yeast is foamy.
In the bowl of a stand mixer fitted with a dough hook attachment, place the flours, salt, ¼ cup of the olive oil, and the yeast mixture and mix on medium speed until the dough forms a ball around the hook, 5 to 6 minutes.
Transfer the dough to a lightly floured surface and knead several times, sprinkling with more flour, as needed, and form into a ball. Brush a large bowl with olive oil, and place the dough inside. Cover with plastic wrap and set aside to rise until doubled in size, 40 to 50 minutes.
Coat a 10 × 15-inch rimmed baking sheet with the remaining ¼ cup olive oil. Punch the dough down, transfer to a lightly floured surface, and knead several times. Place the dough in the pan and press to spread it out to the pan’s edges. Flip the dough over and spread it to the edges again. Make indentations with your fingers, every few inches apart, all over the dough. Cover the baking sheet with plastic wrap and allow the dough to rise until it has doubled in size, about 40 minutes.
Preheat the oven to 425°F. Remove the plastic wrap. Cut the roasted garlic cloves in half and push them into the surface of the dough. Sprinkle with the rosemary, and red pepper flakes, and bake for 20 minutes, until golden brown.