Preheat oven to 400°F (200°C). Line two baking sheets with parchment paper. In separate bowls, toss the beet cubes with 1.5 tbsp olive oil, 1/2 tsp salt, 1/4 tsp pepper, and 1/2 tsp thyme. Toss sweet potato cubes with remaining 1.5 tbsp olive oil, 1/2 tsp salt, 1/4 tsp pepper, and 1/2 tsp thyme.
Spread beets on one baking sheet and sweet potatoes on the other. Roast for 25-30 minutes, stirring halfway through, until tender and lightly caramelized. Remove from oven and let cool slightly.
In a small bowl, whisk together Greek yogurt, extra-virgin olive oil, red wine vinegar, minced garlic, salt, and pepper until smooth. Adjust seasoning to taste.
On a large platter or individual plates, arrange the arugula. Top with roasted beets and sweet potatoes. Drizzle with the yogurt dressing. Sprinkle crumbled feta, toasted walnuts (if using), and fresh dill over the top. Serve immediately.