Melt butter over medium-low heat and steep Earl Grey tea bags in it.
Combine eggs and sugar in a bowl and mix until pale and fluffy.
Sift in cake flour, baking powder, and salt, then mix together.
Add the infused butter with Earl Grey and vanilla extract to the batter and mix.
Let the batter rest in the refrigerator for at least 30 minutes.
Brush butter in the madeleine mold and fill each cavity up to 80% with batter.
Bake in a preheated oven at 180°C (356°F) for 12–15 minutes.