Initial Mixture Preparation: Begin by placing the sweetened condensed milk, peppermint extract, and vanilla extract into the bowl of a stand mixer. Initiate mixing at a low speed, gradually incorporating half of the powdered sugar into the mixture. Continue to slowly add the remaining powdered sugar until a consistent dough is formed.
Dough Preparation: Lay out a piece of parchment paper and evenly distribute approximately 3 tablespoons of powdered sugar on its surface. Remove the dough from the mixer and knead it gently on the sugared parchment paper to ensure thorough blending of ingredients. Proceed to roll the dough into a circle approximately ½ inch in thickness, ensuring a consistent layer of powdered sugar atop the dough to prevent sticking.
Shaping the Patties: Utilize a cookie cutter or the rim of a small glass to cut circles from the dough, aiming for a diameter between 1 and 1 ½ inches. If the dough becomes overly warm and difficult to work with, place it in the freezer for 15 minutes before resuming. Carefully transfer the formed peppermint patties to a parchment-lined baking sheet and place them in the freezer for 15 minutes to firm up.
Chocolate Coating: Melt the dark chocolate in a microwave-safe container, using 30-second intervals for heating and stirring after each interval to prevent the chocolate from burning. Allow the melted chocolate to cool for 3-4 minutes. Employing a fork, dip each peppermint patty into the chocolate, ensuring an even coating on both sides, and then return them to the parchment-lined baking sheet. If desired, garnish with sprinkles at this stage. Refrigerate the chocolate-coated peppermint patties for 20 minutes to solidify before serving.