Preheat oven to 400 degrees F (200 degrees C). Place squash cut-side down in a baking dish; fill with 1/4 inch of water.

Bake squash in the preheated oven until tender, about 45 minutes.

Heat butter in a saucepan over medium heat; cook and stir onion until tender, about 10 minutes.

Heat chicken broth, basil, marjoram, black pepper, and cayenne in a saucepan over medium heat.

Scoop flesh from squash into a blender or food processor; add onion and cream cheese. Blend mixture until smooth, working in batches if needed. Mix squash mixture into broth mixture; cook, stirring frequently, until soup is heated through but not boiling, about 5 minutes.