Preheat the oven to 350 degrees. Chop the chocolate and peppermint sticks. Mix them with the flour and set aside.
In a pot bring 2 cups of water to a low simmer. In a heat resistant bowl, whisk egg whites until foamy. Add sugar to the egg whites. Place the bowl on top of the pot with the water. Whisk the whites constantly until thick and creamy. About 10 minutes.
Remove the bowl from the heat. Whip the whites with an electric mixer until it is thick enough to hold it's shape. Peaks may droop. Fold in the chocolate mixture and peppermint oil with a spatula.
Line 2 sheet pans with parchment. With a large spoon, scoop dollops of meringue onto the pan about 2 inches apart. Bake in the oven for 15-20 minutes. Allow the meringues to cool before removing them from the pan. Store in an airtight container.