Prep the lemongrass: trim off the root end and the dry, grassy top. Peel away the tough outer leaves until you reach the pale, tender core, then chop.
Heat the oil in a large soup pot over medium heat.
Add the onion and lemongrass and cook, stirring occasionally, until soft, about 5–7 minutes.
Add the garlic and ginger and cook until fragrant, about 1 minute.
Stir in the curry paste and cook for 2–3 minutes, stirring constantly so it doesn't stick.
Add the squash, chicken broth, brown sugar, fish sauce, and ¾ of the can of coconut milk. Bring to a boil, then reduce the heat and simmer until the squash is very soft and falls apart when pressed with a spoon, about 30 minutes.
Blend until silky smooth with an immersion blender, or in batches in a regular blender. If you use a regular blender, fill it no more than halfway and remove the center cap from the lid, covering the hole with a folded towel so steam can escape. If the soup is too thick add some water to loosen.
Stir in the lime juice, then season with salt and pepper to taste.
Ladle into bowls and top with chili crisp and a drizzle of the reserved coconut milk.