If desired, dry brine the chicken. Pat the chicken breasts dry. Liberally sprinkle kosher salt all over the surface of the chicken, even on the back. Place the chicken breasts on a plate and cover. Refrigerate for a minimum of 1 hour or up to overnight.
Remove the chicken breasts from the fridge a half-hour before you intend to roast them. Remove any covering and pat the chicken dry. Allow to come up to room temperature uncovered to allow the skin to dry off.
Preheat the oven to 425°F. Line a baking sheet or roasting pan with a rack. Generously drizzle the backs of each breast with olive oil and season with kosher salt and pepper. Repeat on the skin side with more olive oil, kosher salt, and pepper. Place the chicken skin side up on the baking sheet and roast for 35 to 40 minutes or until fully cooked. The chicken is cooked when it reaches an internal temperature of 165°F and the juices run clear when the breast is pierced.
Cut the cabbage in half, leaving the core intact. Cut into 1-inch thick wedges and arrange the cabbage on a baking sheet. Drizzle the wedges with olive oil and season with kosher salt. Flip the wedges and season the other side. After the chicken has been roasting for 10 minutes, begin roasting the cabbage. When the cabbage begins to brown on the bottom, flip the wedges over and continue roasting until the other side browns and the cabbage is tender. Keep warm if the cabbage is done before the chicken.
Peel the outer leaves off the radicchio and cut into wedges. Remove the core and roughly chop. Transfer to a salad bowl or serving platter. Remove the stalk from the kale and roughly chop the leaves. Add to the radicchio and set aside.
When the chicken fully cooked, transfer the breasts to a platter and keep warm. Pour the pan drippings into a measuring cup. Place the baking sheet on the stove over medium heat. Add the shallots and sauté for about one minute until they begin to soften. Add the malt vinegar to the pan and allow it to deglaze the pan, scraping up any bits stuck to the pan. Pour the vinegar along with the shallots into a small mixing bowl.
Add the Dijon mustard, kosher salt, and pepper to the vinegar mixture and whisk together. Measure the pan drippings, adding olive oil as needed to make 1/3 cup. Slowly drizzle the pan drippings into the vinegar, whisking constantly to emulsify the dressing.
Use a sharp knife to remove the breasts from the backbone and cut into thick slices. Remove the cores from the cabbage wedges and roughly chop into bite sized pieces. Add the warm cabbage to the radicchio and kale. Top with about half of the vinaigrette and toss until coated. Add the hazelnuts and toss to combine.
Top each plate with a serving of the salad. Arrange the chicken breast slices alongside the salad. Spoon the remaining vinaigrette over the chicken and enjoy!