Heat a medium soup pot over medium-low heat. Add the EVOO, 3 turns of the pan, the garlic and three-quarters of the chopped onion and cook for 7 to 8 minutes, stirring occasionally. Add the diced tomatoes and the crushed tomatoes and season with salt and pepper. Add the broth and raise the heat until the soup bubbles. Reduce the heat to a simmer and add the bread and beans. Stir the soup as it simmers until it thickens to a stewlike consistency. Turn off the heat, adjust the salt and pepper and ladle the soup into shallow bowls. Top with the grated Parmigiano-Reggiano, an additional drizzle of EVOO and a spoonful of the remaining onion. Garnish with the torn basil. Pass around extra grated Parmigiano-Reggiano at the table.