Heat a gas grill to medium or prepare a medium-hot charcoal fire.

Combine the soy sauce, mirin, honey, rice vinegar, ginger, and garlic in a small saucepan and bring to a boil over medium-high heat. Reduce the heat to medium and gently boil until the mixture is slightly syrupy and reduced to about 2/3 cup, 10 to 15 minutes. Remove from the heat and divide the glaze into two containers.

While the glaze cooks, trim the chicken of excess fat. Cut the chicken thighs crosswise (across the grain) into 1/2-inch wide pieces (about 24 total). Cut the scallions into 2-inch pieces (you'll need 16 pieces). Fold each piece of chicken in half and thread three pieces of chicken onto the skewers, alternating with two scallions and two mushrooms (one of each between the chicken strips).

Brush oil on the grill grates and set the skewers on the hottest part of the grill. Grill until dark grill marks form on the first side, about 6 minutes. Flip the skewers and brush generously with half of the glaze. Continue to cook until the second side is well marked and the chicken is cooked through, another 5 to 6 minutes. Flip the skewers and brush the remaining glaze on the second side. Serve immediately.

Serve with vegetable fried rice or steamed white rice.