Cut a 1/2-inch-thick diagonal slice from a carrot, then roll carrot away from you 90 degrees 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 sauce pan with cider, water, butter bits, 1/4 teaspoon salt, and 1/8 teaspoon pepper, then cover with lid slightly askew and simmer, stirring occasionally, until most of liquid has evaporated and carrots are tender and glazed, about 50 minutes.
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