Preheat the oven to 425°. In a bowl, stir the mustard with the water, mirin, 1 teaspoon of the soy and 1/2 teaspoon of the sugar.
In an ovenproof skillet, heat 1 tablespoon of the oil. Season the chicken with salt and pepper and cook over high heat until golden, 2 minutes. Flip the chicken and brush with the mustard; transfer to the oven and roast for 8 minutes, or until cooked through. Transfer the chicken to a cutting board and let rest for 5 minutes; slice. Wipe out the skillet.
Meanwhile, steam the bok choy for 5 minutes. Drain; pat dry. In the same skillet, heat 1 tablespoon of the oil. Add the bok choy cut side down and cook over high heat, turning once, until browned, 3 minutes. Transfer to a platter; season with salt and pepper. Arrange the chicken over the bok choy.
In a bowl, whisk the vinegar with the remaining oil, soy sauce and sugar; season with salt and pepper. Add the arugula and toss; arrange over the chicken and serve.
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