Use an electric hand mixer or stand mixer on medium speed to beat the butter and sugar until crumbly. Mix in the eggs and vanilla until combined and smooth.
Add half of the flour mixture and half of the milk to the butter mixture, and mix until combined. Add the remaining flour mixture and milk, and mix just until combined or a few small streaks of flour remain.
Divide the batter evenly among the muffin cups, filling each two-thirds full. Bake for 25 to 30 minutes, or until a pick inserted into the center comes out clean.