Make the peppermint ice cream first. (In fact, it's best to do this a day or two ahead of time.) Take the quart of vanilla ice cream out of the freezer and allow it to soften on the counter for about ten minutes. Meanwhile, put the candy canes into a food storage bag. On a plastic cutting board, pound the bag of candy a meat mallet until the canes are broken into tiny pieces.
Scoop about two-thirds of the softened vanilla ice cream into a bowl. Stir the crushed candy canes into the softened ice cream until well distributed. Cover with plastic wrap and put the bowl into the freezer.
When you're ready to make the pie, preheat the oven to 425 degrees. Clean the counter and dust with flour. Unroll the pie dough and lay it out on the counter. Place a nine-inch pie baking pan upside down on top of it, and cut a piece of dough in a circle an inch beyond the edge of the pan. (Use two sheets of pie dough if necessary.)
Grease the pie pan lightly. Push the pie dough into it. Squeeze pleats in the part that extends above the edge, so that it forms an upright collar above the pan. Sprinkle a half-cup of red beans or pie weights into the crust, and bake the crust in a 425-degree oven until light brown. Remove from the oven and allow to cool for a half-hour.
When the crust is cool, remove all the ice creams from the freezer and let them soften a bit in their containers on the counter.
In the meantime, put the egg whites into a clean bowl. Make sure no yolk has gotten into the whites. Add the cream of tartar to the bowl.
Clean the beaters of your electric mixer to remove any fat that may be on them. Beat the egg whites with the bowl tipped a little, with the beaters right in the pool of liquid egg whites at the bottom. Beat until the foam forms soft peaks. Add the sugar and the vanilla and continue beating until the peaks get stiff.
Remove the beans from the cool pie shell. Put the peppermint ice cream into the pie crust, and spread it out to a uniform layer with a rubber spatula. Next, layer in about three-fourths of the chocolate ice cream. (Save the rest for a guilty late-night snack.) Cover the chocolate ice cream with the vanilla ice cream. Finally, spoon the egg-white meringue over the top, forming a high dome of meringue, preferably with a lot of a little peaks.
Put the pie into the 425-degree oven on broil (top heat only). Broil for one or two minutes, until the meringue peaks are distinctly browned and the rest of the meringue lightly brown.