Begin making the dough: place the flour, yeast, olive oil, and 1¼ cups of lukewarm water into an electric mixer fitted with a dough hook. Mix on medium speed for about 5 minutes until a cohesive dough forms. If making by hand, combine the dry ingredients in a large bowl, add the oil and water, and mix with your hands until combined.
Turn the dough out onto a lightly floured surface and knead for 5–10 minutes until it becomes smooth and elastic. Shape the dough into a neat ball and place it in a lightly oiled large bowl.
Cover the bowl with oiled plastic wrap and set it in a warm place to rise for 1–1½ hours, or until the dough has doubled in size and feels puffy to the touch.
Once risen, knock back the dough by punching it down gently to release the air. Turn it out onto a lightly floured surface and knead briefly until smooth again.
Divide the dough into 4 equal pieces. Roll out each piece on a lightly floured surface to form a 9 in (23 cm) round. Preheat the oven to 475°F (240°C).
While the dough is rising, prepare the filling: heat 2 tablespoons of the olive oil in a frying pan over high heat. Add the onions and cook for 1 minute, stirring constantly.
Reduce the heat to low, cover the pan, and let the onions cook gently for 15 minutes until they’re soft and sweet, stirring occasionally to prevent sticking.
Remove the lid and increase the heat back to high. Cook for a few more minutes, stirring frequently, to evaporate any excess liquid until the onions are caramelized and golden.
Transfer the cooked onions to a bowl and stir in the vinegar, mixing well. Set aside while you prepare the other vegetables.
Heat the remaining olive oil in the same frying pan over medium-high heat. Add the peppers and cook for 4 minutes, stirring occasionally, until they begin to soften and char slightly.
Add the mushrooms to the pan with the peppers and continue cooking for 2–3 minutes more, stirring frequently, until both vegetables are tender and any moisture has evaporated.