Preheat the oven to 350°F. Grease a 9x13 cake pan and set aside.
In a medium bowl, whisk the flour, baking powder, baking soda, and salt together. Set aside.
Using a hand mixer or a stand mixer fitted with the paddle attachment, cream the butter with the sugar until smooth. Add the yogurt, milk, and vanilla extract and beat to combine. Add the dry ingredients and mix on low speed until just combined.
Transfer the batter into the prepared cake pan. Bake for 35-38 minutes, or until a toothpick inserted into the center comes out clean. Cool completely before frosting.
To make the chocolate buttercream:
Using a hand mixer or a stand mixer fitted with the paddle attachment, beat the butter until smooth and creamy, about 5-6 minutes.
Add the powdered sugar, cocoa powder, and salt and beat to combine. Add the milk and beat for another 5 minutes.