Cut the carrots into 3-inch logs and then divide each log into halves or fourths, depending on the width of the carrots. This should make spears that have at least one flat side.
Heat 2 tablespoons of the oil in a cast-iron skillet over medium heat. When it is hot, place half of the carrots into the oil flat-side down and cook without turning until tender, 6 to 8 minutes. Sprinkle with salt. The cut sides will be very, very dark. Some may even call it burnt. Don't worry--it tastes great! Stir and let cook until the carrots are tender on the other side, 1 to 2 minutes more. Transfer the carrots to a plate. Repeat the steps to cook the remaining carrots, adding the additional 2 tablespoons oil if needed. Season and cool to room temperature.
Meanwhile, mix together the yogurt, 1/2 teaspoon pepper and a pinch of salt in a small bowl.
Spread the yogurt on the bottom of a serving plate, add the carrots and sprinkle with the walnuts and parsley.