Cut out a round of wax paper to fit just inside a 12-inch heavy skillet, then butter 1 side of round.
Cut a 1/2-inch-thick diagonal slice from a carrot, then roll carrot away from you 90° and cut another 1/2-inch-thick diagonal slice (shape will resemble a trapezoidal log). Continue rolling and cutting carrot, then cut remaining carrots in same manner.
Add carrots to skillet with cider, water, butter bits, 3/4 tsp salt, and 1/4 tsp pepper, then cover with wax-paper round (buttered side down) and simmer, stirring occasionally, until most of liquid has evaporated and carrots are tender and glazed, about 50 minutes.
Servings: 12