Sit the chicken in a large, deep pan. Carefully halve the squash lengthways, then cut into 3cm chunks, discarding the seeds. Slice the coriander stalks, add to the pan with the squash, curry paste and coconut milk, then pour in 1 litre of water. Cover and simmer on a medium heat for 1 hour 20 minutes.
Use tongs to remove the chicken to a platter. Spoon any fat from the surface of the soup over the chicken, then sprinkle with half the coriander leaves. Serve with 2 forks for divvying up the meat at the table. Use a potato masher to crush some of the squash, giving your soup a thicker texture. Taste, season to perfection with sea salt and black pepper, then divide between six bowls and sprinkle with the remaining coriander. Shred and add chicken, as you dig in.