Pat the chicken dry all over and season with ½ teaspoon Kosher salt.
If you’re using bok choy, roughly chop it, keeping the leafy greens separate from the white stalks. If you’re using baby bok choy, which is more tender, simply chop it and set it aside.
Set a deep lidded skillet or Dutch oven over medium-high heat and add the olive oil.
When it’s hot, sear the chicken breasts (in batches if needed to keep from crowding the pan) for 1 to 2 minutes per side, just until they start to brown. Reserve on a plate.
Add the shiitake mushrooms to the pan along with the white bok choy stalks (if using; baby bok choy can be added later). Season with ¼ teaspoon salt and stir-fry for 2 to 3 minutes, until the vegetables are starting to brown along the edges.
Add the ginger, garlic, and pepper (if using) and cook just until fragrant, about 30 seconds.
Add the coconut milk and chicken broth, along with salt to taste (about ½ to ¾ teaspoon). Stir well to combine.
Bring to a simmer, then nestle the chicken breasts inside the pan. Decrease the heat to low and cover.
After 10 minutes, check for doneness by cutting into one chicken breast; if it’s still pink in the center, cover and keep cooking for another few minutes.
A minute or two before it’s done cooking, stir in the bok choy greens and cook just until they wilt.
To serve, slice each chicken breast thinly against the grain. Ladle the veggies and broth into bowls (with rice or quinoa, if using) and arrange the chicken on top.
Finish with a squeeze of lime.
Cut the cucumber in half lengthwise, then slice ¼ inch thick.
Transfer the cucumber to a bowl or colander and toss with the salt. Set aside for 5 to 10 minutes.
Drain any excess liquid from the cucumbers, then toss with the remaining ingredients and serve.
