Cut each squash half into 3 wedges. Arrange in the bottom of a large stockpot with the skin side facing up. Pour 1 cup of the chicken broth over the squash. Bring the broth to a boil, reduce heat to medium-low, place a cover on the stockpot, and cook until the squash is tender, 20 to 30 minutes. Set the squash aside to cool and reserve the liquid. Reserve remaining chicken broth for later.
Melt the butter in a large skillet over medium heat. Cook the onion in the melted butter until softened, about 5 minutes; add the apples and continue cooking and stirring until the apples are tender, 15 to 20 minutes.
Scoop the flesh of the cooled squash into the pitcher of a blender. Pour the reserved cooking liquid over the squash. Blend the squash until smooth. Add the onion and apple mixture and the remaining chicken broth; blend again until smooth.
Pour the blended soup back into the stockpot, and place over medium heat until warmed through.
