Heat vegetable oil in a soup pot over high heat. Add chicken to hot oil; cook, stirring occasionally until browned, about 5 minutes.
Stir in 1/2 of the red onion and brown sugar. Cook until onions have softened and sugar starts to caramelize on the bottom of the pan, about 3 minutes. Stir in black pepper, cumin, and 1 teaspoon kosher salt. Toss in ginger and garlic; stir until sizzling, about 1 minute.
Add 4 cups chicken broth and stir to deglaze the bottom of the pot. Bring to a boil. Add coconut milk and curry paste. Add cayenne pepper and bay leaf. Stir in fish sauce. Reduce heat to medium-low and simmer, stirring occasionally, for 30 minutes.
Carefully cut squash with a large knife on a cutting board. Remove seeds and peel and discard skin. Cut flesh into bite-sized pieces. You should have about 1 1/2 pounds squash.
Add squash to soup and salt to taste. Add remaining chicken broth to cover squash. Simmer over medium-low heat, stirring occasionally, until pumpkin is just barely tender, 15 to 20 minutes.
Stir in remaining red onion, bell pepper, and green onions; cook for 2 minutes. Turn off the heat, and stir in 1/2 cup cilantro and 1/4 cup basil. Taste and adjust seasonings.
Ladle into bowls and garnish with more cilantro, basil, green onions, and serrano chiles. Squeeze lime juice into each bowl.
Don’t use low-fat coconut milk.
The amount of curry paste you use will depend on the brand and strength; it could range from 1 teaspoon to 1/2 cup. You can find curry paste in almost any large grocery store.
You can use any winter squash, like sugar pumpkin, pie pumpkin, kuri, kabocha, hubbard, etc. Do yourself a favor and try to find kabocha–it’s got an extremely pleasant taste with a subtly-sweet flavor, and retains a nice texture, even when fully cooked. Just don’t use a regular pumpkin, because those are very watery and almost tasteless.
For a less brothy soup, don’t add the extra 2 cups chicken broth.
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