Slice the cabbage in half, starting from the top and cutting down through the core. Place each half flat-side down on a cutting board, and slice it thinly around the core to create fine, delicate ribbons. Discard the tough core pieces.
Set a large sauté pan or heavy-bottomed pot over medium-high heat. Pour in the olive oil and add the butter, allowing it to melt completely. Once heated, add the cabbage ribbons, followed by the salt and pepper. Sauté for 10 to 15 minutes, stirring occasionally, until the cabbage softens and starts to turn golden brown. Letting the cabbage rest undisturbed for a few minutes between stirs encourages caramelization, enhancing its flavor.
Remove the pan from the heat and drizzle in the apple cider vinegar. Taste and adjust seasoning as needed, adding a touch more salt, pepper, or a splash of vinegar for extra brightness if desired. Garnish with fresh thyme for an aromatic finish. Serve warm.