Step 1

In small bowl, whisk together miso and 2 tablespoons oil to combine, then whisk in vinegar, soy sauce, and honey. Transfer 3 tablespoons to gallon-size resealable bag, add chicken, and massage to coat; let marinate 30 minutes. Whisk garlic and ginger into remaining miso mixture and set aside.

Step 2

Meanwhile, cook soba per package directions. Drain, rinse, and set aside.

Step 3

Arrange rack 4 to 6 inches from broiler and heat broiler. Line baking sheet with aluminum foil. Arrange marinated chicken, skin side down, on prepared pan (discard marinade). Broil 5 minutes, then flip and cook until browned and cooked through, 8 to 10 minutes more. Let rest 5 minutes before slicing.

Step 4

Meanwhile, heat 2 tablespoons oil in large, high-sided skillet on medium-high. Add mushrooms in even layer and cook undisturbed until starting to brown, 3 to 4 minutes. Toss and continue to cook, tossing once more, until deep golden brown, 3 to 4 minutes more. In last minute of cooking, add scallion whites and cook, stirring occasionally, 1 minute; transfer to plate.

Step 5

Add remaining tablespoon oil to skillet and heat on medium. Add bok choy and cook, stirring often, until leaves are wilted and stems are just tender, 1 to 2 minutes. Fold in baby greens, mushrooms, and soba.

Step 6

Remove from heat and toss with reserved sauce. Serve topped with chicken and dark scallion greens.