Melt 1 tablespoons butter with oil in a heavy large pot over medium-high heat. Add cabbage and onion; saute until vegetables wilt and brown, stirring occasionally, about 15 minutes. Add the thyme sprigs or dried thyme and sauté a minute longer. Add broth and bring to a boil. Reduce heat to medium and simmer 5 minutes. Season to taste with salt and pepper.
Melt remaining 2 tbsps. butter in a heavy large skillet over medium-higih heat. Add apples and saute until brown and tender, stirring occasionally, about 12 minutes. Season with salt and pepper.
Remove thyme sprigs, if used from soup. Ladle soup into bowls; garnish with apples and chopped fresh thyme and serve.