1. For crust: Generously grease a large bowl with olive oil and set aside. Combine water, sugar and yeast in the bowl of a stand mixer fitted with a dough hook attachment. Mix on low speed until combined, about 30 seconds. Let stand until yeast is foamy, about 5 minutes. Add flour and salt. Mix on medium speed until smooth dough is formed, about 3 to 6 minutes. Transfer dough to the prepared bowl and cover with plastic wrap. Let rest until doubled in size, about 1 hour 30 minutes.
2. For topping: Heat a large grill or grill pan to medium-high heat and brush grates with oil. Working in batches, grill eggplant, pepper and red onion, turning occasionally, until lightly charred and tender, about 5 to 7 minutes. Cut peppers into 1/4-inch slices.
3. On a lightly floured surface, press dough into a 14-inch round circle. Transfer dough round to lightly floured pizza peel. Quickly slide dough round onto grill. Grill until char marks appear, about 2 to 3 minutes. Flip dough round over and top with cheese, grilled vegetables and mushrooms, leaving a 1/2-inch border. Continue to grill, covered, until crust is golden-brown and cheese is melted, about 5 to 8 minutes. Top with basil. Slice and serve with Chardonnay.