Preheat oven to 350°F. Grease an 8-inch square baking pan.
In the prepared pan, add 1½ cups all-purpose flour, 3 Tbsp unsweetened cocoa powder, 1 cup sugar, 1 tsp baking soda, and ½ tsp salt. Stir to combine and level the dry mixture.
Make three small wells in the dry mixture. Pour 1 tsp white vinegar into the first well, 1 tsp vanilla into the second well, and 5 Tbsp vegetable oil into the third well.
Pour 1 cup water evenly over the entire mixture. Stir directly in the pan with a spatula or whisk until the batter is smooth and no dry streaks remain, scraping the sides and corners.
Bake 35–40 minutes, until a toothpick inserted in the center comes out clean or with a few moist crumbs. Remove from oven and allow the cake to cool completely in the pan.
To make the frosting, place ½ cup chocolate chips and 1 Tbsp butter in a medium saucepan over medium-low heat. Stir constantly until the chocolate and butter are fully melted and smooth, then remove from heat.
Stir in ½ tsp vanilla. Add ½ cup powdered sugar and mix; if the frosting is too thin, add more powdered sugar up to ¾ cup total until you reach a spreadable consistency. If the frosting is too thick, stir in the optional 1 Tbsp milk, a little at a time, until desired consistency.
Spread the frosting over the completely cooled cake. Let the frosting set slightly, then slice and serve.