Heat a soup pot over medium-high heat. Add the EVOO (3 turns of the pan), then add the pancetta and cook for 4 minutes. Add the garlic, three-quarters of the chopped onion, the carrots and zucchini and season with salt and pepper. Cook the veggies for 7 to 8 minutes, then add the wine to deglaze the pot. Stir in the tomatoes and stock and bring up the heat. When the soup boils, reduce it to a simmer and stir in the bread and beans. Pile the greens into the pot and wilt them into the soup.
Simmer the ribollita for 5 to 10 minutes, stirring the soup as it simmers, until it thickens to a dense stewlike consistency. Turn off the heat, adjust the seasonings and ladle into shallow bowls. Top each bowl with some of the grated Parmigiano-Reggiano cheese, an additional drizzle of EVOO, a spoonful of the reserved raw onion and some basil. Pass around the remaining cheese at the table.