Sprinkle meat with cheese and mushrooms. Top with remaining tomato sauce. Roll up meat jelly roll fashion, starting at narrow end. Use waxed paper to help. Place meat seam side down.
Seal ends securely.
Bake at 375 degrees for 30 minutes. Drain off any fat that accumulates. Return to oven for 30 minutes. Mix in cream of rice. Cut carrots into quarters lengthwise and in half crosswise.