Heat vegetable oil in a large soup pot over medium-high heat.
Add chicken pieces and cook until no longer pink. Set aside.
In the same pot, sauté chopped onion and minced garlic until fragrant.
Stir in Thai red curry paste and cook for another minute.
Add pumpkin puree, coconut milk, and chicken broth. Mix well.
Bring to a gentle simmer and cook for 15-20 minutes.
Optionally, use an immersion blender for a smoother texture.
Return cooked chicken to the pot. Add diced red bell pepper and carrots.
Include fish sauce and brown sugar if desired. Season with salt and pepper.
Simmer for an additional 10-15 minutes until vegetables are tender and chicken is cooked through.
Serve hot, garnished with fresh cilantro leaves and lime wedges if desired.