Pat chicken dry and season all over with salt. Arrange, skin side down, in a cold, dry medium Dutch oven and set over medium heat. Cook, undisturbed, until skin is golden brown and crisp and easily releases from pot, 8–10 minutes. Slide thighs around to different spots (this will ensure even browning) and continue to cook until deeply browned, 3–5 minutes longer. Transfer chicken to a plate. Let pot cool 2 minutes.
Set pot over medium-low heat; add shallots, garlic, and ginger and cook, stirring constantly, until very fragrant but not browned, about 2 minutes.
Stir in rice and star anise and cook, stirring constantly, 1 minute.
Stir in soy sauce, ¾ tsp. salt, and 3 cups water.
Nestle chicken back into rice mixture in pot, arranging skin side up. Increase heat to medium and bring liquid to a simmer. Immediately cover pot, reduce heat to low, and cook 25 minutes. Remove from heat and let chicken and rice sit 10 minutes.
While the chicken and rice are cooking, make the sauce. Whisk peanut butter, soy sauce, vinegar, Sriracha, ginger, honey, and ¼ cup warm water in a small bowl until smooth and pourable. If sauce still looks a little thick, continuing adding water until it reaches a drizzle-able consistency.
Remove lid from pot and fluff rice with a fork. Pluck out and discard star anise. Serve chicken and rice with peanut sauce,cucumber, and cilantro alongside.