In a medium stainless steel bowl, beat the egg white until frothy but not stiff. (DO NOT USE a plastic bowl.) Slowly add the powdered sugar while blending with an electric mixer set on medium speed. Add the corn syrup and peppermint oil or extract and knead the mixture with your hands until it has the smooth consistency of dough. Add more powdered sugar if necessary, until mixture is no longer sticky.
Using a surface and rolling pin heavily dusted with cornstarch, roll out the peppermint dough until it is about 1/4 inch thick. Punch out circles of peppermint dough with a biscuit cutter or a clean can with a diameter of about 2 1/2 inches. Make approximately 20, place them on plates or cookie sheets, and let them firm up in the refrigerator, about 45 minutes.
Melt the chocolate chips in a microwave set on high for 2 minutes. Stir halfway through the heating time. Melt thoroughly, but do not overheat. Drop each patty into the chocolate and coat completely. Using a large serving fork, or 2 dinner forks, one in each hand, lift the coated patty from the chocolate. Gently tap the forks against the bowl to knock off the excess chocolate, and place each patty on wax paper. Chill the peppermint patties until firm, about 30 minutes.