Generously grease a madeleine pan with butter, making sure to coat each cavity well.

Add the milk and butter to a small saucepan. Heat over low, stirring occasionally, just until the butter has completely melted. Do not let the mixture boil. Remove from heat.

In a large bowl, whisk together the egg, sugar, salt, and vanilla extract until the sugar has mostly dissolved. Slowly pour in the warm milk and butter mixture while whisking continuously until smooth.

Add the glutinous rice flour and tapioca starch. Whisk until the batter is completely smooth and no dry pockets remain.

Pour the batter through a fine-mesh sieve into a clean bowl or large measuring cup. Let the batter rest for 5 to 10 minutes.

Preheat oven to 365°F. Fill each cavity of the prepared madeleine pan about three-quarters full. Bake for 35-40 minutes, or until the tops are deep golden brown and the edges are crisp.

Let the butter tteok cool in the pan for about 5 to 10 minutes, then transfer to a wire rack to cool. Serve and enjoy!