Preheat a charcoal or gas grill to medium indirect heat.
Pierce sunchokes 4 to 5 times with a fork. Grill sunchokes and squash, turning frequently, until tender and lightly charred, about 20 to 40 minutes. Once tender, remove from grill and allow to come to room temperature. Cut the sunchokes and squash into bite-sized pieces and season lightly with salt. Reserve.
For the vinaigrette, combine miso, vinegar and honey in a blender. With the blender on medium-high speed, slowly stream the oil into the mixture and blend until emulsified. Add salt and pepper to taste.
Season the grilled vegetables and lettuces with salt and pepper, then divide evenly between six plates. Drizzle a tablespoon or two of the vinaigrette over the dish and season with a pinch of finishing salt. Serve.
*Osaka finishing salt is a blend of sesame seeds, Sancho pepper, sugar, dried shiso, and dried seaweed. It can be purchased at worldspice.com.
**South River Three-Year Barley Miso is preferred.