Preheat the oven to 375°. Arrange the chicken breasts on a parchment-lined baking sheet. Rub the chicken with olive oil and season with salt and pepper. Sprinkle the thyme over the chicken. Roast in the oven for 40 minutes, or until the skin is crisp and the chicken is cooked through.
Remove to a cutting board and rest for 5 minutes, then sliced 1/2-inch thick on a bias.
Meanwhile, in a large saucepan, cover the rice with the water, add 1 teaspoon of salt and bring to a boil. Cover and simmer over moderately low heat until the wild rice is tender but still chewy, about 1 hour. Drain the wild rice well.
In a small skillet, toast the almonds over medium heat until golden brown.
Toss the wild rice with the butter, parsley and almonds. Season with salt and transfer to a platter. Arrange the chicken on top of the wild rice and serve warm.