In a large skillet stir together the broth, carrot, and onion. Bring to boiling; reduce heat. Simmer, covered, for 5 minutes. Stir in bulgur and cranberries. Remove from heat. Cover and let stand for 5 minutes. Drain off excess liquid.
Meanwhile, halve the sweet peppers lengthwise, removing the seeds and membranes.
Stir shredded cheese into bulgur mixture; spoon into sweet pepper halves, dividing evenly. Place sweet pepper halves in skillet. Add the 1/2 cup water. Bring to boiling; reduce heat. Simmer, covered, for 5 to 10 minutes or until sweet peppers are crisp-tender and bulgur mixture is heated through. Sprinkle with nuts.