Preheat Oven:
Preheat oven to 350°F (175°C). Line baking sheets with parchment paper.
Mix Dry Ingredients:
In a medium bowl, whisk together flour, baking powder, baking soda, and salt. Set aside.
Cream Butter and Sugar:
In a large bowl, beat together butter, oil, and sugar until light and fluffy.
Add Eggs and Vanilla:
Beat in eggs one at a time, then stir in vanilla extract.
Combine Wet and Dry Ingredients:
Alternately add flour mixture and sour cream/milk to the butter mixture, beginning and ending with flour. Mix until smooth.
Shape Cookies:
Using a large spoon or ice cream scoop, drop batter onto prepared baking sheets, spacing about 2 inches apart. Flatten slightly to form rounds.
Bake:
Bake for 15–18 minutes, until edges are lightly golden. Cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.
Prepare Glazes:
For vanilla glaze, whisk powdered sugar, milk, and vanilla until smooth.
For chocolate glaze, whisk powdered sugar, cocoa, milk, and vanilla until smooth.
Ice the Cookies:
Spread vanilla glaze on one half of each cookie and chocolate glaze on the other half. Allow glaze to set before serving.