Preheat oven to 450°F with rack in lower third position.

Whisk ¼ cup each oil and lemon juice, 1 tablespoon dill, the grated garlic, 1½ teaspoons mustard and ¼ teaspoon pepper in a medium bowl until combined. Set aside ¼ cup of the mixture in a small bowl.

Add 4 chicken thighs to the medium bowl with the oil-lemon mixture and turn to coat; let stand for 15 minutes while the oven preheats.

While the chicken marinates, cut cabbage into 8 wedges, keeping the core intact.

Place the cabbage wedges and 3 cups sweet potatoes on a large rimmed baking sheet. Drizzle with the remaining 2 tablespoons oil; toss to coat.

Arrange the cabbage wedges, cut-side down, and the sweet potatoes in an even layer.

Place the chicken, skin-side up, in the corners of the baking sheet, resting on top of the sweet potatoes if necessary; sprinkle with ¼ teaspoon salt.

Roast until the cabbage is tender and browned in spots and an instant-read thermometer inserted into the thickest portion of chicken registers at least 165°F, about 40 minutes, flipping cabbage halfway through.

Sprinkle ¼ teaspoon salt over the cabbage and sweet potatoes; toss to coat.

Meanwhile, stir yogurt and the remaining 2 tablespoons dill and ¼ teaspoon salt into the reserved oil-lemon mixture (if necessary, stir in water, 1 teaspoon at a time, to reach a thick drizzling consistency).

Divide the chicken and vegetables among 4 plates; drizzle with the yogurt mixture. Garnish with additional dill, if desired.