Melt 1/4 cup unsalted butter in a large saucepan over medium heat until foamy.
Cook and stir 1 cup chopped onion until golden brown, about 10 minutes.
Stir in 2 cans crushed tomatoes, 1 tablespoon crushed fennel seed, 4 teaspoons salt, and 1/4 teaspoon ground black pepper.
Bring the mixture to a boil, then reduce heat and simmer to blend the flavors, about 30 minutes.
Stir in 2 cups half-and-half and 1 cup chopped basil, and heat on low until the soup is hot but not boiling.
Puree the bisque until smooth using an immersion blender or a regular blender in batches, filling the pitcher no more than halfway full.
Hold down the lid of the blender with a folded kitchen towel and carefully start the blender using a few quick pulses before leaving it on to puree.