Beat the egg white until frothy but not stiff. (There's going to be a lot of beating, so a stand mixer is a little better than a hand mixer for this recipe.
Add 4 cups of the powdered sugar, beating at about medium speed. You'll have a crumbly mix at this stage, not quite a dough.
Add the corn syrup and peppermint extract, and stir either by hand or with the paddle attachment until combined. Add the remaining cup of powdered sugar, working it in about 1/4 cup at a time. By now it should be the texture of a cookie dough. Add more sugar if the mix is still sticky.
Heavily dust a countertop with powdered sugar, then turn out the dough and roll until about 1/4 inch thick. Using a cookie cutter, biscuit cutter, or even a shot glass, cut rounds out of the dough. Knead leftover dough into a ball and repeat until all has been used up.
Place the rounds on a cookie sheet lined with parchment paper and chill in the refrigerator to firm them up, about 45 minutes.
Melt the chocolate with the oil either in the microwave or over a double boiler.
Using a fork, dip the patties in the chocolate one by one, then return to the cookie sheet.
Chill in the refrigerator until the chocolate is set and the patties are firm, about 30 minutes.