Preheat oven to 350 F. Grease and flour a 9x13-inch baking dish. Set aside.
In a medium bowl, whisk together flour, baking soda, and salt. Set aside.
In a large bowl, beat butter and sugar together until combined.
Beat in eggs, buttermilk, and vanilla until well-combined.
Add mashed potatoes and melted chocolate. Mix until incorporated.
Stir in flour mixture until just incorporated.
Pour batter into prepared baking pan, smoothing as needed.
Bake for 35-50 minutes, or until a wooden toothpick inserted in the center comes out clean.
Let cake cool to room temperature before frosting.
To make the frosting
Beat butter and powdered sugar together until well-combined.
Mix in chocolate and milk until completely incorporated.
Spread over cooled cake.
Notes
Cake flour: If you don't have cake flour, you can make a homemade substitute with this recipe. Also, to properly measure the flour, stir or sift it to break it up, lightly spoon it into the measuring cup, and level.
Butter: I use salted butter in this recipe. You can use unsalted if you prefer. "Cool room temperature" means that the butter is about 68F. The butter should dent when pressed, but the stick shouldn't lose its shape or be smooshy.
Buttermilk: You can use regular or low-fat buttermilk. If you don't have buttermilk, you can make a substitute with this recipe.
Mashed potatoes: Instant or regular will work. If you don't have leftover mashed potatoes, I have a section above the recipe that describes how to make enough mashed potatoes for this cake.