Make the dough: In a large bowl, add the yeast and ½ cup warm water (105–110°F); stir to combine. Set aside until the surface is very foamy, 5–10 minutes.
On a large, clean work surface, add the flour and salt, stir to combine, and form into a tall mound with a deep well in the center. Add 1½ cups cool water to the yeast mixture, then pour into the well. Using a fork, slowly mix the liquid and the flour a little at a time, moving in short, counterclockwise circles around the border of the well and pulling in a bit of flour with every turn, until a shaggy dough forms. When the dough is firm enough to hold its shape, use a dough scraper or wide spatula to scrape the remaining flour over the mass. Knead the dough until smooth and shiny, 7–10 minutes, then grease a large bowl with a drizzle of oil and transfer the dough to the bowl. Turn the dough over a few times to coat in oil, then cover the bowl loosely with plastic wrap and set aside at room temperature until nearly doubled in size, about 2 hours.
Punch down the dough, fold it in thirds, then lift and flip it over completely. Cover loosely with plastic and set aside at room temperature until very gassy and nearly doubled in size, 35–40 minutes more.
Meanwhile, make the chile oil: In a small pot over low heat, add ¼ cup oil. Once the oil is hot and shimmering but not yet smoking, add the chile flakes, then turn off the heat, cover, and set aside to cool to room temperature.
Turn the dough out onto a lightly floured work surface. Divide into 4 even pieces (about 10 ounces each) and roll each piece into a loose ball. Cover loosely with plastic wrap and let rest while you preheat the grill.
Set up a charcoal grill for indirect cooking, or a gas grill for two-zone cooking (alternatively, preheat a grill pan over high heat). Set a small bowl of oil and a pastry brush by the grill or stove.
Liberally oil a clean work surface. Working with one piece of dough at a time, use your fingers to dimple and stretch the dough out to a thin, even rectangle, about 16 by 9 inches.
Grill the dough over direct heat (or in the hot grill pan) until the dough is evenly charred on the bottom and very bubbly and beginning to set on the top, 2–3 minutes. Using tongs, flip the dough, then brush the charred side lightly with oil. Continue cooking until the opposite side is crispy and charred in places, about 2 minutes more, then flip again and brush lightly with oil. Transfer the dough to indirect heat (or turn the heat under the grill pan down to medium), then top with ¼ of the cheese, followed by ¼ of the peaches and prosciutto. Continue cooking just until the cheese and prosciutto fat are softened and beginning to melt, and the peaches are warmed, 2–3 minutes more. Slide the pizza onto a cutting board, then top with a generous handful of fresh arugula and 1 tablespoon of the chile oil. Repeat with the remaining dough and toppings. Cut the pizzas into strips and serve immediately.