Preheat oven to 400 degrees F (200 degrees C). Place buttercup squash halves cut side-down in 2 casserole dishes. Add 1 1/2 inches water.
Bake in the preheated oven until skin is easily pierced with a knife, about 1 hour. Drain off water and cool squash until easily handled.
Scoop buttercup squash flesh into a large bowl. Add 1/4 cup butter and 1/2 teaspoon salt; mash until smooth.
Combine water, chicken breasts, 1 onion, chicken bouillon, celery leaves, salt, and pepper in a large pot. Bring to a boil; reduce heat and simmer until chicken breasts are no longer pink in the center, 30 to 45 minutes.
Remove chicken from the pot and cut into bite-size chunks. Strain broth into a bowl; discard onion and celery leaves.
Heat 1/4 cup butter and olive oil in a large pot over medium-low heat. Add onion and garlic; cook and sit until translucent, about 5 minutes. Add mashed squash and chicken broth; simmer until flavors combine, about 20 minutes. Stir in chicken, potatoes, and corn; simmer until potatoes are tender, about 20 minutes.
Heat half-and-half in a small saucepan until warmed through, 5 to 10 minutes. Stir into the pot slowly. Cook soup over low heat until flavors combine, about 15 minutes.
This recipe can be made simpler if you use premade chicken broth and prepare the chicken or squash in advance. Whatever you do, don't pour cold half-and-half into the soup!
My mother uses a lot of butter. Omit the butter in the squash if desired.
Carrots are also a nice touch in this soup. Omit the chicken if you're not a meat eater!
Substitute sweet potatoes for the potatoes if desired.
You will need at least 1 casserole dish for every buttercup squash. They should lie flat with the cut side down. Use an additional dish if both halves will not fit in 1 dish.