Preheat an oven to 400 degrees F (200 degrees C). Toss the yams with 2 tablespoons of olive oil in a bowl. Season to taste with salt and pepper, and arrange evenly onto a baking sheet.
Bake in the preheated oven until the yams are tender, 20 to 25 minutes. Cool to room temperature in the refrigerator.
Meanwhile, heat the remaining 1 tablespoon of olive oil in a large skillet over medium heat. Cook and stir the onion and garlic until the onion has caramelized to a golden brown, about 15 minutes. Stir in the kale, cooking until wilted and tender. Transfer the kale mixture to a bowl, and cool to room temperature in the refrigerator.
Once all the ingredients have cooled, combine the yams, kale, red wine vinegar, and fresh thyme in a bowl. Season to taste with salt and pepper, and gently stir to combine.