Sift the flour, baking powder, baking soda, and salt together in a medium mixing bowl and set aside.
In a stand mixer or large mixing bowl, add the butter, shortening, and sour cream (all at room temperature) and cream together until nice and smooth. This will take about 2-3 minutes. Add the sugar, egg, and vanilla, and peppermint extracts and mix until combined.
Add dry ingredients to the wet ingredients in 3 batches, mixing in completely after each addition.
Form a loose ball with the dough and place it in a covered bowl in the fridge for an hour.
Preheat the oven to 400°F and line your baking sheets with a silicone mat or parchment paper.
Once the dough has chilled, scoop out 2 tablespoons of the dough and form a ball with your hands. Place the ball on the baking sheet and repeat until the sheet is full (you will probably only be able to fit 12-14 balls).
Sprinkle the tops of the cookie balls with powdered sugar. This will keep the glass from sticking to the dough. You can also dip the bottom of the glass into the powdered sugar as well. Flatten each ball until it's about ¼-1/2 inch thick.
Bake the cookies one pan at a time for 10 minutes. The cookies will spread and puff up a little but won't turn brown, so be careful not to overbake them.
Remove the cookies from the oven and allow them to cool on the pan for 5 minutes before placing them on a cooling rack to cool completely.