Mix wet ingredients: In the bowl of a stand mixer, whisk together warm water, olive oil, and vinegar by hand. Fit mixer with the paddle attachment.

Add dry ingredients: In a separate bowl, whisk together flour, salt, yeast, and sugar. With mixer on low speed, gradually add dry ingredients to wet mixture. Beat until combined, about 1 minute. Increase speed to medium and beat for 5 minutes, stopping to scrape paddle and bowl. Dough will be very soft but should hold together.

Rest dough: Cover dough and let stand for 20 minutes.

Shape rolls: Spray a rimmed baking sheet with cooking spray. Heavily dust a work surface with flour. Turn dough out, flour top, and gently pat into a 9x6-inch rectangle. Cut into 6 (3-inch) squares. Place on prepared pan. Cover and let rise in a warm, draft-free spot (75°F/24°C) until puffed, about 30 minutes.

Preheat oven: Position rack in top third of oven and preheat to 500°F (260°C).

Bake: Bake until lightly browned and the center registers at least 210°F (99°C), 20–25 minutes. Let cool on pan for 10 minutes, then transfer to a wire rack to cool completely. Serve with herbed olive oil if desired.