Heat vegetable oil in a large, heavy-bottomed pot over medium-high heat. Season chicken pieces with salt and pepper, then brown them on all sides until golden, about 8-10 minutes total. Remove chicken and set aside.

In the same pot, sauté onions until translucent, about 5 minutes. Add garlic and cook for another minute until fragrant.

Return chicken to the pot and add chicken stock or water, ensuring liquid covers the chicken by about 2 inches. Bring to a boil, then reduce heat and simmer for 30 minutes.

Add yam chunks, carrots, thyme sprigs, and allspice to the pot. Continue simmering for 20-25 minutes until yams are tender and chicken is falling off the bone.

Carefully add the scotch bonnet pepper, starting with just a small amount and tasting as you go. Remember, you can always add more heat, but you can't take it away.

Stir in coconut milk and simmer for an additional 10 minutes, being careful not to let the soup boil vigorously as this may cause the coconut milk to curdle.

Taste and adjust seasoning with salt, pepper, and additional scotch bonnet if desired. Remove thyme sprigs.

Stir in chopped green onions, cilantro, and lime juice just before serving. Ladle into bowls and serve immediately while hot.