Special equipment: a 4-inch-diameter cookie cutter and 12 paper lollipop sticks (4- or 6-inch sticks)
Position oven racks in the top and bottom thirds of the oven and preheat to 225 degrees F. Cut 2 pieces of parchment paper to fit 2 baking sheets. Using a 4-inch diameter cookie cutter, draw 6 circles on each piece of parchment, then flip each piece over onto a baking sheet marked-side down.
Combine the cream of tartar, salt and egg whites in the bowl of a stand mixer fitted with the whisk attachment. Whisk on medium speed until foamy, about 1 minute. Increase the speed to medium high and beat until thickened, about 1 minute more. Mix in the peppermint extract, then add the confectioners' sugar in batches about 1/4 cup at a time, beating until well incorporated after each addition. After the last addition, continue to beat until stiff, shiny peaks form, about 5 minutes.
Fill a resealable plastic bag or piping bag with the meringue. Snip off the end or corner to make about a 1/2-inch-wide opening. Lay a lollipop stick down about 1 inch into each drawn circle and pipe the meringue lollipops onto the parchment circles over the sticks. Sprinkle the sanding sugar and silver sprinkles onto the piped meringue pops.
Bake, rotating the baking sheets from front to back and top to bottom halfway through, until the meringue is dry and crispy, about 2 hours.
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