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.
