In a large bowl, use a hand or stand mixer to gradually blend the sugar into the cream cheese until smooth, scraping the sides as needed. Add the peppermint and blend to incorporate.
Line two baking sheets with waxed or parchment paper. Scoop a level teaspoon of dough, then roll it into a ball and place it on a prepared sheet. Repeat with the remaining dough, spacing the balls 1/2 inch apart on the sheets.
Dust a 1/4-teaspoon measuring spoon with confectioners' sugar, then use it to indent the center of each mint. If needed, dust the spoon again as you work. Chill the mints in the refrigerator until firm, about 2 hours.
Melt the chocolate according to the package directions. Transfer it to a piping bag or a ziplock bag with one corner snipped off. Fill the center of each mint with chocolate, then return the mints to the refrigerator until the chocolate sets, about 45 minutes. Keep the mints refrigerated when not being served; because they contain cream cheese, they shouldn't sit out more than a couple of hours.