Preheat the oven to 425 degrees . In a medium bowl, whisk the hoisin, lime juice, 1 tbsp. oil, the vinegar, ginger and garlic; season with salt and pepper. Reserve half the glaze in a small bowl. Place the chicken in a large cast-iron skillet. Using kitchen twine, tie the legs together; brush the chicken with the remaining glaze.
Roast the chicken 45 minutes, brushing with the reserved glaze every 15 minutes (tent with foil if browning too quickly).
In two separate bowls, toss the mushrooms and bok choy with 1 1/2 tsp. oil each; season. Arrange the mushrooms around the chicken and roast 5 minutes more. Arrange the bok choy and scallions around the chicken; roast until an instant-read thermometer inserted into the thickest part of the thigh registers 165 degrees , about 10 minutes more. Sprinkle the chicken and vegetables with sesame seeds.