Melt the butter using a microwave or stovetop and set it aside to cool slightly so it is not too hot when added to the batter.

In a large bowl, whisk together the flour, sugar, baking powder, baking soda, and salt until evenly combined.

In another bowl, whisk the eggs, buttermilk, and vanilla extract until smooth, then slowly mix in the melted butter.

Pour the wet ingredients into the dry ingredients and gently whisk until just combined. The batter should be thick with a few lumps remaining.

Heat a griddle or skillet over medium heat and grease it with butter or nonstick spray.

Scoop or pour a portion of batter onto the hot surface and cook until the edges look set and bubbles form on the surface.

Flip the pancake and cook until the other side is golden and cooked through.

Repeat with the remaining batter, greasing the pan as needed, and keep the cooked pancakes warm in a low-temperature oven until ready to serve.