Preheat the oven to 350 degrees F (175 degrees C). Line 2 baking sheets with parchment paper.
Combine sugar and butter in the bowl of a stand mixer; beat until light and fluffy. Beat in Greek yogurt, eggs, and vanilla extract. Add all-purpose flour, whole wheat flour, baking powder, and baking soda; mix until combined. Fold in sprinkles.
Scoop out even amounts of dough using a cookie scoop and roll into balls. Arrange on the prepared baking sheets. Use your hand or a glass to flatten the balls.
Bake in the preheated oven until edges are lightly golden, 10 to 12 minutes. Transfer to a cooking rack and repeat with remaining dough. Allow cookies to cool completely, about 15 minutes.
Make frosting while the cookies are cooling. Beat butter until creamy; mix in confectioners' sugar. Stir in vanilla extract. Thin frosting with cream if needed.
Spread frosting over cookies using a knife. Cover with more sprinkles.