In a bowl, whisk together the dry ingredients: flour, baking powder, and salt. Set aside.
With a mixer, beat butter and sugar until combined. Add the eggs and blend well.
Alternate adding the sour cream and flour mixture until the flour is completely incorporated. The dough will be very sticky. Wrap dough in plastic wrap and refrigerate for one hour or longer.
Place dough on a lightly floured surface and dust the top with flour. Also dust the rolling pin with flour.
Using the rolling pin, roll the dough to ½ inch thickness. Using a round cookie cutter, cut dough into circles and use a smaller cutter to create the center holw. Knead any scrap pieces of dough into a mound of dough and continue rolling out and cutting the doughnuts until all of the dough is used.
Heat oil in a dutch oven or deep fryer between 325 and 350 degrees.
Carefully place doughnuts in oil a few at a time. Do not crowd the oil. Cook until golden brown, about 2 minutes on each side.
Remove doughnuts and place on a wire cooling rack or paper towel.