Preheat the oven to 400 degrees F. Bring a pot of water to a boil, add the rice and cook until tender, 15 to 20 minutes. Drain; rinse under cold water.
Meanwhile, toss the carrots with the cinnamon, 1 tablespoon vinegar, and salt and pepper to taste in a bowl. Sprinkle the pork with 1/4 teaspoon each salt and pepper. Heat the olive oil in a large ovenproof skillet over medium-high heat. Add the pork and brown all over, 5 minutes. Add the carrots and cook 2 minutes. Transfer the skillet to the oven; roast until a thermometer inserted into the pork registers 150 degrees F, 8 to 10 minutes. Transfer the pork to a cutting board.
Whisk the yogurt, pecans, garlic, the remaining 1 tablespoon vinegar, 1 tablespoon water, 1/4 teaspoon salt, and pepper to taste in a large bowl. Stir in the roasted carrots and pan drippings. Add the greens, dried cranberries and rice and toss. Season with salt and pepper and divide among plates. Slice the pork and add to the salads.