Digital kitchen scale, measuring spoons, chef's knife, 2 clean kitchen towels, 11-inch straight-sided saute pan or large skillet, digital instant-read thermometer, stand mixer with paddle attachment, rubber spatula, rolling pin, 2-inch round cutter, 2 half sheet pans, parchment paper or 2 silicone baking mats, pastry brush (if needed), medium to large metal bowl for melting chocolate, a fork
Place a folded kitchen towel in the bottom of an 11-inch straight-sided saute pan or large skillet. Add 1-inch of water and place over medium-high heat and bring to a bare simmer, about 190 degrees F. (Yes, there's a towel in the pan…it's okay.)
Combine 30 grams (about 1/4 cup) of the confectioners' sugar with the egg white in the bowl of your stand mixer, then park the bowl on the towel in the water. Tilt the bowl as needed so the egg mixture is in contact in the water. (Some mixers have an added piece of metal at the bottom that necessitates this.) Reduce the heat to medium and stir often with a rubber spatula until the egg mixture is smooth, foamy, and hits 160 degrees F, 7 to 10 minutes. Remove from the water and transfer bowl to the mixer. Kill the heat under the saute pan but leave it on the stove; you'll use it later.
Add the corn syrup and mint oil to the egg mixture. Using the paddle attachment, beat on medium-high until combined, about 30 seconds, scraping down the bowl as needed. Reduce the speed to low, add one-quarter of the confectioners' sugar, and beat until combined, 30 to 45 seconds. Continue to add the sugar in this manner until it's all incorporated, then continue beating until the mixture forms a dough, about 1 minute. Remove the paddle attachment and knead a few times with your hands to smooth out the dough and form it into a ball. If the dough does not form a ball, add 1/2 teaspoon of water, return the bowl to the mixer, and beat with the paddle until smooth.
Lightly dust the counter and a rolling pin with the cornstarch. Transfer the dough to the counter and roll into a rectangle about 1/4-inch-thick. Use a 2-inch ring cutter to stamp out 25 to 30 rounds of dough. Place the rounds on a parchment-lined half sheet pan, then re-roll and stamp the scraps. You should have about 3 dozen rounds of dough. If there is a lot of excess cornstarch remaining on the rounds, brush off with a pastry brush.
Place the pan with the peppermint rounds near the cooktop and position a second pan next to it lined with either a silicone baking mat or a second sheet of parchment. Place a folded kitchen towel right next to the stovetop.
Turn the heat under the saute pan back on to medium-high and let the water return to a bare simmer. Reduce the heat as low as it will go.
Place the chocolate in a metal bowl that will fit in the saute pan and set in the hot water. Stir constantly for 5 seconds. Remove from the heat, set on the towel, and stir vigorously for 30 seconds. Return to the hot water and stir for another 5 seconds, then back to the dry towel for 30 seconds. Repeat until the chocolate is half-melted, then reduce the time the bowl is on the heat from 5 to 3 seconds. Keep the off-heat stir time at 30 seconds. The goal is to melt all the chocolate while keeping the temperature below 91 degrees F. This process can take anywhere from 10 to 20 minutes, depending on the size of the chocolate pieces. If the chocolate begins to creep up above 91 degrees F, leave the bowl on the counter and stir like crazy until the temperature drops back down. Continue with the melting process.
Once the chocolate is a smooth liquid, stir vigorously for another minute before dipping. Place a candy round into the bowl of chocolate and, using a fork, flip it over to coat both sides. Pull the candy from the chocolate, then tap the fork on the side of the bowl to remove as much excess chocolate as possible. Transfer to the silicone mat and repeat with the remaining candies. (If the chocolate becomes too thick or hard to work with, return it to the hot water for a few seconds, stirring constantly. Again, do not let the temperature rise above 91 degrees F.)
Rest the candies at room temperature until the chocolate hardens, about 5 minutes. Devour or store in an airtight container at room temperature for up to 2 months.