Place squash, cut-side down, in a baking pan and add 1-inch of water. Cover pan with aluminum foil.
Bake squash in the preheated oven until tender, about 1 hour. Cool until able to handle, about 15 minutes. Scoop flesh into a bowl and discard squash skin.
Melt butter in a large pot over medium heat. Cook and stir onion, celery, poblano pepper, and jalapeno pepper in hot butter until softened but not browned, about 15 minutes.
Stir chicken broth, cooked squash, and salt into onion mixture; bring to a boil. Reduce heat to low and simmer until flavors blend, about 30 minutes.
Blend soup with an immersion blender or in batches in a blender until smooth. Stir brown sugar into soup; simmer for 10 minutes. Adjust brown sugar to taste.
