Makes 8 servings or 1 (9-inch) cake
Make the cake: Preheat the oven to 350°F. Sift together the flour, cocoa powder, baking powder, baking soda, and salt. In a separate bowl, combine the sugar, butter, water, lemon juice, and vanilla. Combine the wet and dry ingredients, scraping the bowl with a plastic spatula to be sure all ingredients are fully incorporated. Whisk the batter until just combined, making sure to not overmix.
Pour the batter into a greased and floured 9-inch round cake pan lined with a circle of parchment paper on the bottom. Bake for 30 minutes, or until the cake springs back to the touch and a toothpick inserted in the center comes out clean. Remove the cake and let it cool on a wire rack.
Meanwhile, make the frosting. In a microwave-safe bowl, heat the chocolate chips in 30 second intervals, stirring in between. Be sure not to overheat or the chocolate will seize up. Set aside.
While the melted chocolate cools, begin whipping the butter. Place the butter in a mixing bowl and whip until it turns pale yellow and fluffy. Add the powdered sugar 1 cup at a time, being sure to mix long enough to get the consistency back to fluffy. Scrape the sides of the bowl between additions to incorporate all of the butter. Add the vanilla and half of the cooled chocolate. Whisk until ingredients are fully incorporated and fluffy. Add the remaining chocolate and combine, scraping sides of the bowl. Depending on how fluffy you want your frosting, add 1 tablespoon of milk at a time until the frosting reaches the desired fluffiness.
Spread the frosting in an even layer across the top of the cake and serve.
For the cake:
1½ cups all-purpose flour, plus more for dusting
¼ cup unsweetened cocoa powder, sifted
1 teaspoon baking powder
¾ teaspoon baking soda
¼ teaspoon kosher salt
1 cup granulated sugar
⅓ cup unsalted butter, melted, plus more for greasing
1 cup warm water
1 tablespoon lemon juice (or vinegar)
1 teaspoon vanilla extract
For the frosting:
1½ cups semi-sweet chocolate chips
1½ cups unsalted butter, room temperature
3 cups powdered sugar
3 teaspoons vanilla extract
1 to 3 tablespoons whole milk (optional)