In a medium bowl, stir together the flour, yeast, salt, and sugar. Add the water and using a wooden spoon or your hand (I find using my hand easier), mix until you have a wet, sticky dough, about 30 seconds. Lightly coat a second medium bowl with olive oil and place dough in it. Cover and let sit at room temperature until the dough is more than doubled and the surface is dotted with bubbles, about 8-9 hours. (This is perfect for putting together the night before and letting rise while you sleep.)
When the first rise is done, generously dust a board or work surface and scrap the dough out of the bowl in one piece. Using floured hands or a bench scraper, fold the dough over itself two or three times and nudge it into a loose flat ball. Brush or spray the surface with olive oil and sprinkle a pinch of sugar over the top. Let sit, uncovered in a warm draft-free spot and allow to rise until doubled, about 1-2 hours.
A half-hour before the end of second rise, preheat oven to 500°F and put rack in center of oven. Oil a sided cookie pan or half sheet pan. Transfer the dough to the pan in one piece, and spread out to a relatively even thickness. It doesn’t necessarily have to go all the way into the corners but should be even thickness so it cooks uniformly. Using your fingers (or if you have nails, your knuckles so you don’t tear the dough) make dimples all over the dough. Push the pieces of peach into the dimples, then brush with olive oil (or a little melted butter) and sprinkle with 1-2 teaspoons sugar.
Bake in the center of the oven for 12-15 minutes, rotating pan halfway through. Remove from pan to a cooling rack and let cool about 15 minutes. Slice up and enjoy with a cappuccino! Calories: approximately 185 per slice for 8 slices, 125 for 12.