Arrange oven racks in the center and upper third positions. Preheat the oven to 425°F.
Put the shredded cabbage on a large baking sheet. Add a drizzle of extra-virgin olive oil, and a sprinkling of kosher salt and freshly-ground black pepper. Toss to coat. Then spread the cabbage out in an even layer. Set aside.
Peel the sweet potato, and then slice it into 1/4-inch coins or "medallions." Arrange the medallions in a single layer on a small (or large) baking sheet. Brush the tops with olive oil and sprinkle with kosher salt and freshly-ground black pepper. Set aside.
Pat the chicken thighs dry with paper towels. For each thigh, gently loosen the skin, and then place 2 (or more) sage leaves between skin and meat. Dust the skin with kosher salt and freshly-ground black pepper.
Warm the butter and olive oil in a skillet over medium heat. When the butter melts, add the chicken thighs skin-side-down. Sauté just until the skin turns chestnut brown -- 90 seconds to 2 minutes. Immediately remove from heat.
Make 4 openings in the cabbage bed, and add a chicken thigh to each opening. Place this cabbage-and-chicken baking sheet on the center rack of the preheated oven; put the other baking sheet that holds the sweet potato medallions on the upper third rack.
Bake until the cabbage turns brown and crisp around the edges, the chicken registers 165°F, and the sweet potato medallions are caramelized and tender -- usually 25 minutes.