Quarter carrots lengthwise, then cut into 2 1/2-inch pieces.
Heat butter in a 12-inch heavy skillet over moderately high heat until foam subsides, then sauté carrots, uncovered, stirring occasionally, until slightly softened, about 5 minutes.
Add brown sugar, stirring until sugar is melted. Stir in water, lemon juice, salt, cinnamon, and cayenne and simmer, covered, until carrots are tender and liquid is reduced to a glaze, 8 to 10 minutes.
