Heat oil in a large pot over medium heat.
Add onions, garlic, celery, parsley, lovage and salt.
Cook, stirring, until onions are translucent, about 5 minutes.
Stir in tomatoes and stock; bring to a boil.
Reduce heat; simmer until tomatoes are soft and mixture is slightly reduced, 15 to 20 minutes.
Stir in basil; cook until fragrant, about 5 minutes.
Pass soup through the medium plate of a food mill set over a large bowl; discard solids.
Serve with crostini.