Butter and flour an 8-inch-square baking tin. Preheat the oven to 350 degrees F.
Place the eggs into an 8-ounce (1-cup) measuring cup. Fill the rest of the cup up with milk to the 1-cup line. Pour the eggy milk into a large bowl and beat. Then beat in the sugar. Sift together the flour and baking powder and fold it into the eggnog mixture.
Melt the chocolate and butter in a double boiler over very low heat. Cool slightly, then fold it into the cake batter with the vanilla.
Pour the batter into the prepared pan and bake for 35 to 40 minutes. That's it. You can frost it (when it's cool, of course) or just eat it plain.