Sift together the flour, baking powder, and salt into a large bowl and set aside.
In a separate large bowl or stand mixer fitted with a paddle attachment, cream together the butter and sugar at high speed until light and fluffy, about 2 minutes.
Add the eggs one at a time, scraping down the sides of the bowl as needed.
Add the vanilla and half of the sour cream and mix until combined.
Add the dry ingredients to the wet ingredients slowly, mixing just until incorporated.
Mix in the rest of the sour cream at low speed just until combined, do not overmix.
Flour a large piece of plastic wrap and scoop the dough out onto it. It might be easier to split the dough in half and use two pieces. Refrigerate the dough for 1 hour.
Preheat the oven to 325°F and line a baking sheet with parchment paper and set aside.
Remove the chilled dough from the refrigerator and dust a large flat surface with flour.
Divide the dough into thirds and use a rolling pin to roll out a portion of the dough until it's 1/4 inch thick. Use a 3-inch round cookie cutter to cut out the cookies and place them on the prepared cookie sheet about 2 inches apart. These cookies don't spread much. Repeat with the remaining portions of dough.
Bake for about 10 to 12 minutes, just until edges start to brown, remove from the oven and transfer to a cooling rack to cool completely. Bake off the remaining cookies.