Heat 1 ½ tablespoons of oil in a large (11 inches or larger), deep skillet over medium-high heat. Meanwhile, season the chicken on both sides with 1 teaspoon of kosher salt.

Gently swirl the oil in the pan to coat the bottom of the skillet, then add the chicken breasts in a single layer. If you have a splatter guard, use it. Let cook on one side, until golden brown and the chicken releases from the pan, 4 to 6 minutes. Use tongs to flip the chicken and cook until browned on the other side, about 4 minutes.

Transfer the chicken to a plate and cover with foil—it will not be cooked through. Lower the heat to medium; do not wipe out the pan.

Add the remaining ½ tablespoon of oil to the pan followed by the shallots and ¾ teaspoon salt and sauté until beginning to turn translucent, 1 to 2 minutes.

Add the green beans and garlic and toss until fragrant, about 1 minute. Add the ginger and stir, then add the broth. Scrape the bottom of the pan to release any stuck-on bits.

Add the coconut cream and stir to combine. Add the chicken back to the pan, nestling it among the green beans so it is as submerged as possible. Bring to a simmer and cook, turning the chicken occasionally and stirring the sauce, until the chicken registers 165°F in the thickest parts, 7 to 10 minutes.

Turn off the heat. Add the lime zest and the lime juice and stir. Taste, adding more juice and/or salt as needed. Top with fresh cilantro and serve sprinkled with coconut and lime wedges on the side, if desired, along with rice.