Melt ¼ cup of milk chocolate chips according to the instructions on the bag. Set aside.
Preheat the oven to 350°F (176°C) and grease a 9-inch round cake pan with butter and cocoa powder. Line the bottom with parchment paper and grease that also.
In a medium bowl, whisk together the flour, baking soda and salt.
In a large bowl, cream the butter and sugars until light and fluffy, and then cream in the instant pudding mix.
Beat in the egg and yolk, making sure they are fully incorporated into the batter. Mix in the melted chocolate and vanilla extract.
On low speed, mix in half of the flour mixture at a time, until just combined. Fold in the chocolate chips and spread the batter evenly in the prepared pan.
Bake for 24 to 26 minutes. The middle should be soft but not doughy. Let the cake cool completely in the pan. The cake will firm up more as it cools.
Once cool, carefully flip the cake out onto a cooling rack. (It will be upside down.) Then flip the cake right-side up onto a cake board or serving platter.