Big hats might be rather practical, if you're a farmer and have to endure long times in hot sun and nothing to hide ... especially in Italy. The Mexican Sombrero has a similar idea.
But agreed, it doesn't look like a peasant's hat. But if I'd seen other peasants with such hats, I would believe it, no problem. But possibly it's the boss of the peasant's, who just looks, if everybody works well. And the sun is indeed hot, but luckily he has enough money to pay for the big hat.