Slice the chicken breasts from the bone and discard the skin; reserve the bones. Season the breast meat generously on both sides with salt and 1 teaspoon of the hawaij, put on a plate, and refrigerate. Remove the drumsticks, thighs, and wings from the carcass.

Warm the oil in a large pot over medium heat. Add the onion, garlic, and a punch of salt and cook, stirring frequently, until the onion begins to soften, about 10 minutes. Add the chicken stock and the chicken pieces (except for the reserved breast meat), breast bones and carcass. Raise the heat to medium-high and bring to a boil, skimming off any foam that rises to the surface. Lower the heat to a simmer and add 1 tablespoon of the hawaij. Simmer for about 45 minutes, until the chicken is cooked through. Transfer the thighs and drumsticks to a plate, cover and refrigerate. Simmer the soup for another 2 hours. Pour the soup through a fine mesh strainer into a clean soup pot; discard the solids.

Place the pot over medium heat and add the reserved chicken thighs and drumsticks, along with the potatoes and remaining 1 teaspoon of hawaij. Bring to a simmer and cook until the potatoes are tender, about 20 minutes. Add the reserved chicken breasts and ramps or scallions and simmer until the chicken is just cooked through, about 10 minutes more. Remove the chicken breasts from the soup and gently pull the meat apart using two forks. Season the broth with salt and more hawaij if you like.

To serve, transfer the chicken thighs and drumsticks and potatoes to a platter. Divide the shredded chicken and ramps or scallions among four soup bowls, ladle in the broth, and serve with the platter of chicken and potatoes.