Put the chicken into a large pot, cover it with water, and add a big pinch of salt. Cover and bring it to a boil over high heat. Reduce to a simmer, skim the scum, and cook until the chicken is cooked through, 35 to 40 minutes. Remove the chicken. Strain the broth and put it back into the pot over medium heat.
When the chicken is cool enough to handle, shred the meat and discard the skin and bones. Add the coconut milk, chile, scallion, ginger, lime juice, and chicken to the broth and simmer for 10 minutes. Taste and adjust the seasoning with salt and pepper. Serve in bowls garnished with cilantro.