If the cooked rice is cold, reheat it in the microwave. When hot, mix in the Cheddar until the cheese melts. Set aside to cool to room temperature.
Mash the avocado and lime juice together in a small bowl with a fork until it becomes a spreadable consistency.
Lay the tortilla out on a cutting board. Spread the mashed avocado on the tortilla, leaving a 1-inch outer border without any avocado. In the bottom third of the tortilla, lay out in a log shape the rice, beans, carrots and salsa. Roll over once tightly, then tuck in the sides and continue rolling. The avocado should help the burrito stay together. Cut in half. Wrap in a paper towel, followed by a layer of aluminum foil. Serve at room temperature.