Pat the chicken breasts dry and season all over with 1/2 teaspoon kosher salt.
If using regular bok choy, roughly chop it and keep the leafy greens separate from the white stalks; if using baby bok choy, chop it and set aside.
Heat the olive oil in a deep lidded skillet or Dutch oven over medium-high heat until shimmering.
Sear the chicken breasts in the hot pan, working in batches if needed, 1 to 2 minutes per side until lightly browned; transfer to a plate.
Add the mushrooms and the white bok choy stalks to the pan (add baby bok choy now if using), season with 1/4 teaspoon salt, and stir-fry 2 to 3 minutes until edges begin to brown.
Add the minced ginger, garlic, and the hot pepper or red pepper flakes (if using) and cook about 30 seconds until fragrant.
Pour in the coconut milk and chicken broth, add additional salt to taste (about 1/2 to 3/4 teaspoon total), and bring to a gentle simmer.
Nestle the seared chicken breasts into the simmering liquid, reduce the heat to low, cover, and simmer for about 10 minutes. Check one breast for doneness and cook a few more minutes if still pink in the center.
One to two minutes before the chicken is done, stir in the bok choy greens and cook just until wilted.
While the chicken cooks, halve the cucumber lengthwise and slice into 1/4-inch pieces; toss the cucumber with 1/2 teaspoon kosher salt in a bowl or colander and let sit 5 to 10 minutes, then drain any excess liquid.
Toss the drained cucumber with 1 1/2 teaspoons lime juice or rice vinegar, 1 teaspoon sesame oil, 1 teaspoon black sesame seeds, and 1/4 teaspoon red pepper flakes.
To serve, thinly slice the chicken against the grain. Ladle the mushrooms, bok choy, and coconut broth into bowls over rice or quinoa if using, arrange the sliced chicken on top, and finish with a squeeze of lime.