Prepare the Brussels sprouts:

1. Preheat oven to 425°F. Trim the stem ends and remove any damaged outer leaves. Halve the sprouts lengthwise, quartering particularly large ones.
2. Toss with 2 tbsp olive oil, salt, pepper, and thyme. Spread on a large baking sheet, cut sides down.

Begin roasting:

1. Roast the Brussels sprouts for 12–15 minutes, until the cut surfaces begin to brown.

Prepare the apples:

1. While the sprouts roast, core the apples and cut into thick wedges, approximately 3/4 inch.
2. Leave the skins on for color and texture.
3. Toss with the remaining 1/2 tbsp olive oil and the optional smoked paprika.

Add the apples:

1. Remove the baking sheet and gently distribute the apple wedges between the Brussels sprouts.
2. Return to the oven for another 12–15 minutes, until the sprouts are deeply golden and the apples are tender but still hold their shape.

Toast the walnuts:

1. While everything finishes roasting, toast the walnuts in a dry skillet over medium-low heat for 3–5 minutes, stirring frequently.
2. Remove from heat once fragrant.
3. Prepare the finishing dressing:
4. In a small bowl, whisk together the apple cider vinegar and optional Dijon mustard.
5. If using butter, melt it and whisk it into the dressing.

Finish and serve:

1. Transfer the roasted sprouts and apples to a serving bowl.
2. Toss lightly with the dressing, sprinkle with the toasted walnuts, and adjust salt and pepper to taste.
3. Serve warm.