In a large heavy pot, heat olive oil over medium-high. Add onion and carrots; sauté until softened and translucent (about 8–10 min). Season with dried basil, 1 tsp salt and ½ tsp pepper as they cook.
Stir in tomatoes (with purée), and chicken stock. Cover and simmer on medium-low for about 30 minutes, until vegetables are very soft.
Purée in batches in a blender (carefully!) or with an immersion blender until completely smooth.
Stir in heavy cream and remaining 2½ tsp salt and remaining ½ tsp pepper. Taste it! Add more salt + pepper as needed - until the flavor pops.
Ladle into bowls, top with basil ribbons, and serve with crostini.
