In a large bowl, whisk together the flour, cocoa powder, baking soda, and salt. Set aside.
Using a hand mixer or stand mixer, beat the butter, granulated sugar, and brown sugar until light and creamy, about 3 minutes.
Add the egg and vanilla extract and mix until combined.
Gradually add the dry ingredients and mix on low speed until a thick dough forms.
Line baking sheets with parchment paper.
Roll dough into small balls using about 2 teaspoons of dough for each cookie.
Place on prepared baking sheets and flatten each ball into a 1.5-inch disc using the bottom of a glass or measuring cup.
Refrigerate the shaped cookies for 10 minutes.
Preheat the oven to 350°F (177°C).
Arrange chilled cookies about 3 inches apart on baking sheets.
Bake for 8–10 minutes until edges are set and tops begin to crack slightly.
Cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.
To make the filling, beat the butter and shortening together until creamy.
Add confectioners' sugar and vanilla extract. Beat until smooth and fluffy.
Spread or pipe the filling onto half of the cooled cookies.
Top with remaining cookies and gently press together to create Homemade Oreos.