Step 1Heat the oven to 200 degrees. Grease 2 rimmed baking sheets with butter and set aside.
Step 2In a large, heavy-bottom saucepan, combine the sugar, corn syrup and water and bring to a boil over medium-high heat. Continue cooking until a candy thermometer reaches 290 degrees. Remove the pan from heat and, when the bubbles subside, carefully stir the peppermint extract into the sugar mixture.
Step 3Pour four-fifths of the mixture onto one prepared sheet and place it into the heated oven. Pour the remaining sugar mixture onto the other prepared sheet and add a dozen or so drops of food coloring to the sugar.
Step 4Using a greased bench scraper or metal spatula, spread out the mixture and fold it over itself repeatedly until the coloring is evenly distributed and the sugar is cool enough to handle without burning your hands through sugar gloves.
Step 5Wearing greased sugar gloves, begin "pulling" the candy by stretching it out into a long strip, folding it over and pulling it again. After 10 to 15 minutes, when the candy is lightened and opaque in color and is almost too tough to continue pulling, stretch the strip out once more so it is about one-half inch thick. Divide the strip into 4 equal sections and place them back on the second baking sheet and into the oven.
Step 6Remove the first baking sheet, and work this larger batch of sugar mixture in the same manner as the smaller, pulling the sugar until firm and opaque in color and forming it into a strip about 2 inches thick. Divide this strip into 4 equal sections, and place all but one section in the oven.
Step 7Take 1 of the red sugar strips from the oven and line it up with the thicker white strip, pressing the strips together to form one log. Roll the log out until it is about one-half inch in diameter, and carefully twist the log to give it the familiar candy cane look. Cut the log into desired lengths, and twist the top of each length to form a hook. If at any time the log becomes too difficult to work, place it in the oven until it becomes soft enough to manipulate. Repeat, twisting together remaining pairs of strips.
Step 8Set the candy canes out onto a piece of parchment paper to harden, then store the candy canes in an airtight container in a cool, dry place.