Melt the chocolate over a double boiler or in the microwave until smooth. Set aside to cool slightly while you prepare the batter.
Preheat the oven to 350°F. Weigh the bowl of your stand mixer and write that down for later. Line a cupcake pan with paper cases.
In a small bowl, whisk together the eggs, milk, and vanilla. Set aside. In the bowl of a stand mixer, sift together the flour, sugar, baking powder, and salt. Mix the dry ingredients on the lowest setting until combined, about 30 seconds.
With the stand mixer still on the lowest speed, add the butter one piece at a time, waiting a few seconds between additions. Mix until the butter and flour begin to clump together, about 30-40 seconds after the last piece of butter is added. Scrape the bowl down.
Add 3/4 of the egg mixture with the mixer on low and beat for 5- 10 seconds to incorporate. Bump the speed up to medium- high and beat until light and fluffy, about a minute. Scrape the bowl down.
Add the remaining quarter of the egg mixture in a slow, steady stream down the edge of the bowl. If you hit the paddle, it will make a mess. It should take about 30 seconds to add this remaining mixture. Stop the stand mixer to scrape down the sides of the bowl. Beat again at medium- high speed for about 15 seconds to thoroughly combine.
Split the cake batter in two with the help of an electric scale. Subtract the weight of the bowl (from step 2) from the weight of the batter in the bowl. Divide that number in half. That is the amount of batter you will remove and place in a second bowl. Add the melted and cooled chocolate to one of the bowls. Gently fold the chocolate into the batter until it is combined.
Alternate tablespoon scoops of vanilla and chocolate batter in each cupcake cavity. Each will take about 3- 4 tablespoons. Wiggle a skewer through the batter.
Bake for 20- 22 minutes or until the cupcakes spring back when touched. A skewer should come out clean.
Cool in the pans for about 5- 10 minutes before transferring to a cooling rack. Cupcakes must cool completely before assembling.