Heat olive oil in a large saucepan over medium heat. Add garlic and red pepper flakes. Cook, stirring occasionally, until garlic is lightly browned, about 4 minutes.
Stir in crushed tomatoes, bread, chicken broth, basil sprig, salt, and pepper. Bring the mixture to a simmer, then reduce heat to medium-low. Cover and cook, stirring occasionally, until the bread is softened and the soup thickens slightly, about 15 minutes.
Remove and discard the basil sprig. Whisk the soup until the bread breaks down completely, and the soup reaches a thick, smooth consistency, about 1 minute.
Garnish each portion with grated Parmesan, chopped basil, and a drizzle of extra olive oil. Serve hot.