Preheat the oven to 350 degrees. Line a muffin pan with six liners.
Melt the butter and stir the sugar very well.
Next, stir in the egg white and vanilla. Stir until very well mixed.
Sprinkle the flour, baking powder, baking soda and salt evenly over the mixture.
Stir to combine while slowly adding the milk.
Finally, stir in the sprinkles.
Divide the batter among the cupcake liners and bake for 15-17 minutes, until the tops of the cupcakes are set and spring back when lightly touched.
Let the cupcakes cool completely, leaving them in the muffin pan.
While the cupcakes are cooling, make the frosting. Beat together the butter and powdered sugar. Only add the milk if the frosting needs help coming together into a light, fluffy mixture. Stir in the sprinkles right before frosting the cupcakes and serving them.