Step 1

In a large saucepan over medium, heat the oil until shimmering. Add the onion, celery, pepper flakes and ½ teaspoon salt. Cook, stirring, until the vegetables are softened and lightly browned, 6 to 7 minutes. Stir in the tomatoes with juices, 4 cups water, the Parmesan rind (if using), 1 teaspoon salt and ½ teaspoon black pepper. Bring to a simmer over medium-high, then reduce to medium-low and cook, uncovered and stirring occasionally, until darkened and slightly thickened, about 1 hour.

Step 2

Meanwhile, heat the broiler with a rack about 6 inches from the element. Generously brush both sides of each bread slice with oil and place in a single layer on a broiler-safe rimmed baking sheet. Broil until golden brown on both sides, 2 to 3 minutes, flipping halfway through and rotating the baking sheet as needed. Remove from the oven and lightly rub the garlic clove over both sides of each toast. Tear the toasts into 1-inch pieces and divide evenly among 4 serving bowls; set aside.

Step 3

When the soup is done, remove and discard the Parmesan rind (if using). Taste and season with salt and black pepper. With the soup simmering over medium, break 1 egg into a small bowl, ramekin or teacup. Add the egg to the soup by holding the bowl just above the surface, tilting it and letting the egg slide in. Working quickly, repeat with the remaining eggs, spacing them evenly and noting the order added. Simmer gently, covered and without stirring, until the whites are set but the yolks are still soft, about 6 minutes; do not boil.

Step 4

Using a slotted spoon, remove the eggs in the order they were added, allowing excess liquid to drain off; place one in each serving bowl. Ladle the soup around the eggs. Drizzle with additional oil and sprinkle with basil and Parmesan.