Heat a medium Dutch oven over medium-high heat. Add the olive oil and pancetta, and cook until the pancetta is brown and crispy, about 4 minutes. Remove to a paper-towel-lined plate to drain.
Add the red onion, carrot, garlic and red pepper flakes to the pot. Reduce the heat to medium and cook, stirring often, until fragrant, 1 minute. Add the salt, canned tomatoes, chicken broth and bread cubes; stir to combine. Nestle in the basil sprigs and Parmesan rind. Bring to a simmer, then reduce the heat to maintain a gentle bubble. Stir in the pancetta.
Cook for 25 minutes, stirring often to break apart the bread and prevent the Parmesan rind from sticking to the bottom of the pan. If the soup gets too thick, add extra broth as needed; the finished soup should be silky and thickened. Stir in the grated Parmesan.
To serve, ladle the soup into bowls and top with a sprinkling of Parmesan, a drizzle of olive oil, and chopped basil.