In a food processor, pulse the gochujang, mirin, sake, vinegar, oil, garlic, and ginger until smooth. Reserve 1 cup marinade, and chill.
Add chicken to a large glass baking dish or bowl, and pour remaining marinade (3/4 cup) over chicken. Toss to coat. Cover with plastic, and marinate in the refrigerator at least 2 hours or overnight.
Let chicken come to room temperature, about 30 minutes. Remove from marinade, and season with salt and pepper.
Heat grill to medium-high, and oil grates. Brush bell peppers, zucchini, and summer squash with oil. Grill, flipping halfway, until lightly charred in places and crisp tender, about 3 minutes per side. Transfer to a shallow dish, and season with salt and pepper. Pour half the reserved marinade over top, and let flavors meld.
Grill chicken, basting occasionally with reserved marinade, until lightly charred in places and cooked through, about 6 minutes per side. Transfer to a platter with vegetables, and serve warm.
Recipe by Laura Rege.