Whisk flour, sugar, baking powder, and salt together in a bowl. Beat eggs in a separate bowl using an electric mixer for 1 minute; add beer, milk, oil, and vanilla extract and beat on low for 30 seconds. Pour egg mixture into flour mixture and mix just until combined.
Heat a lightly oiled griddle to 400 degrees F (200 degrees C). Drop 1/2 cup batter onto the griddle and cook until bubbles form and the edges are dry, 3 to 4 minutes. Flip and cook until browned on the other side, 2 to 3 minutes. Repeat with remaining batter.