Set a large deep skillet or a dutch oven over medium heat, add oil. Once oil is hot, add chicken and cook until lightly golden brown, about 4 minutes.
Add ginger, green onions, and garlic, saute for 2 minutes, then add sweet potatoes and curry paste. Cook for 2 minutes or until curry paste is fragrant.
Add fish sauce, brown sugar, lime zest and juice, chicken broth, salt, and basil Turn heat up to medium-high and bring to a low boil. Reduce to a simmer, cook until potatoes and squash are fork-tender, about 20-25 minutes.
Whisk in coconut milk, continue cooking for 5 minutes, then remove from the heat.
Serve warm, garnish with fresh cilantro and peanuts, if desired.