Dice the shallot, mince the garlic, grate the fresh ginger, chop the cilantro, and squeeze the fresh lime juice.
Place the chicken breasts between two sheets of parchment paper and gently pound them to an even thickness.
Season both sides generously with salt and freshly ground black pepper.
Heat the avocado oil in a large skillet over medium heat.
Once the oil is hot, add the chicken in a single layer.
Cook undisturbed for 3 to 4 minutes until a golden crust forms.
Flip the chicken and cook another 3 to 4 minutes until mostly cooked through.
Transfer the chicken to a plate and reduce the heat to medium-low.
Add the diced shallot and minced garlic to the skillet. Cook for 1 to 2 minutes until softened and fragrant.
Pour in the chicken broth, coconut milk, lime juice, grated ginger, and chopped cilantro.
Stir well while scraping up any browned bits from the bottom of the skillet to build extra flavor.
Return the chicken to the pan, nestling it into the creamy sauce.
Cover and simmer over low heat for about 5 minutes, or until the chicken reaches an internal temperature of 165°F (74°C).
Spoon the creamy coconut lime sauce generously over the chicken before serving.
Garnish with fresh cilantro and lime slices if desired.