Place the softened butter into a small bowl. Add the granulated sugar and mix with a small rubber spatula until smooth.
Add in the egg yolk. Mix until fully incorporated with the spatula.
Add in the vanilla extract and whole milk. Mix until just combined.
Add in the flour. Mix until just combined.
Add in the sprinkles. Mix until just combined. Scoop batter into an 8 oz microwave-safe coffee mug and top with a pinch of extra sprinkles.
Microwave on high for 1 minute and 5 seconds. Allow cake to cool while preparing the glaze.
Prepare the glaze by combining the powdered sugar and whole milk. Mix until smooth. Pour over the warm birthday cake. Top with more sprinkles.
Tips
Home microwaves vary in how they cook. You may need to adjust the time up or down. If in doubt, the cake must cook for at least one minute. Add additional time in 5-second increments until the cake is puffed and cooked through.