Mince and mash garlic to a paste with a pinch of salt.
Boil soy sauce, Sherry, and brown sugar in a small saucepan over medium heat, stirring occasionally, until reduced to about 2/3 cup, 5 to 7 minutes. Stir in garlic paste, ginger, and sesame oil, then cool to room temperature.
Pour half of marinade into a large sealable bag and chill remainder for basting. Add chicken to bag and marinate, chilled, turning bag occasionally, at least 3 hours.
Prepare grill for direct-heat cooking over medium-hot charcoal (medium-high heat for gas).
Meanwhile, halve bok choy lengthwise and blanch in a large pot of boiling salted water (2 Tbsp salt for 5 qt water) until crisp-tender, about 2 minutes. Immediately transfer with tongs to an ice bath to stop cooking. Pat bok choy very dry, then, bending leaves, thread 3 or 4 halves (through bulb and leaves) onto each of 3 or 4 skewers. Put on a tray.
Toss mushrooms with vegetable oil. Thread mushrooms (through sides) and chicken (discard marinade) onto remaining skewers, alternating them. Put on another tray.
Oil grill rack, then grill chicken-and-mushroom skewers, covered only if using a gas grill, 6 minutes. Turn over and baste with some of remaining marinade, then grill, turning and basting occasionally (but not during last 3 minutes), until chicken is just cooked through and mushrooms are tender, 6 to 8 minutes more.
Lightly brush bok choy with oil and grill, covered only if using a gas grill, turning once, until grill marks appear, about 2 minutes total.