The skillet will be very full at the start of step 3, but the cabbage will shrink as it cooks.

Adjust oven rack to middle position and heat oven to 375 degrees. Cook bacon in 12-inch nonstick skillet over medium heat, stirring often, until bacon is crispy and fat is rendered, 5 to 7 minutes. Using slotted spoon, transfer bacon to paper towel–lined plate; set aside.

Pat chicken dry with paper towels and sprinkle all over with 1 teaspoon salt and pepper. Reheat fat left in skillet over medium-high heat until shimmering. Place chicken in pan, skin side down, and cook until skin is well browned, 6 to 9 minutes. Transfer chicken to plate, skin side up.

Add cabbage, potatoes, garlic, and remaining 1 teaspoon salt to fat left in skillet and cook, without stirring, until cabbage begins to wilt, 2 to 4 minutes. Gently stir cabbage mixture and continue to cook, stirring occasionally, until cabbage begins to brown, about 2 minutes longer.

Stir in broth, mustard, and caraway seeds. Nestle chicken into cabbage mixture, skin side up, and bring broth to simmer. Transfer skillet to oven and cook, uncovered, until chicken registers at least 185 degrees, 30 to 35 minutes.

Transfer chicken to clean plate (skillet handle will be hot). Stir vinegar and 1 tablespoon dill into cabbage mixture in skillet. Divide cabbage mixture among 4 shallow serving bowls and top each portion with 2 chicken thighs. Sprinkle with reserved bacon and remaining 1 tablespoon dill. Serve.

Cutting the Cabbage