Heat oil in a large pot over medium-high; cook chicken 5-6 minutes until lightly browned and cooked through and set aside.
Sauté garlic, ginger, and curry paste in same pot 1-2 minutes until fragrant, stirring constantly to prevent burning.
Stir in broth and coconut milk; bring to a gentle simmer.
Add mushrooms and pepper; simmer 8-10 minutes until tender.
Stir in fish sauce, lime juice, and sugar; return chicken and heat through 2-3 minutes.
Adjust seasoning to taste; add fresh cilantro just before serving.
Ladle into bowls; garnish with green onions and serve hot with extra lime wedges.