In a large bowl, whisk together the eggs, sugar, and vanilla until well combined.
Gradually add the milk, whisking until smooth.
Sift the flour into a separate bowl, then slowly pour in the wet ingredients, whisking continuously to create a lump-free batter.
Stir in the cooled, melted butter, strain to get rid of any possible lumps, cover, and refrigerate for at least 30 minutes.
Heat a lightly buttered pan over medium heat. Pour about 1/3 cup of batter into the center, tilting and swirling the pan to spread it into a thin, even layer.
Cook for 1–2 minutes, until the edges lift slightly and the bottom is golden brown. Flip and cook for another 30 seconds to 1 minute on the other side.
Repeat with the remaining batter, lightly greasing the pan as needed.