Preheat the oven to 350 degrees. Grease two small (half pint) canning jars.
In a small mixing bowl, whisk together the egg and sugar until it's lighter in color and a bit frothy. Add the butter, vanilla, and milk and whisk for another 30 seconds or so. Add the flour, baking powder, and pinch salt and whisk until just combined. Add the sprinkles and gently fold in with a spatula or spoon. Don't go crazy with the stirring here, or they'll run all over the place.
Fill each jar about 1/2 full and place on a baking sheet. Bake for about 20 minutes or until a toothpick inserted in the center of the jar comes out clean. Cool completely before frosting.
For the frosting, beat together the butter, vanilla, and milk until smooth. Slowly add in the powdered sugar, about a 1/4 cup at a time, beating well after each addition. Pipe or spread the frosting onto the cakes.