Melt the butter in a saucepan or microwave and let it cool slightly. Grind the Earl Grey tea leaves as finely as possible.
In a large bowl, whisk together the sugar and eggs until pale and slightly thickened. Then, add the milk and vanilla extract and whisk until combined.
In a separate bowl, whisk together the flour, baking powder, salt, and ground Earl Grey tea. Make sure the tea is evenly distributed throughout the flour mixture.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Be careful not to overmix; a few lumps are okay.
Gently fold in the cooled melted butter and lemon zest until everything is well incorporated.
Cover the bowl with plastic wrap and refrigerate for at least 30 minutes, or up to 2 hours. This step is crucial for achieving the characteristic madeleine hump.
Preheat your oven to 375°F (190°C). Grease a madeleine pan thoroughly with butter and dust with flour, tapping out any excess. This will prevent the madeleines from sticking.
Spoon or pipe the batter into the madeleine molds, filling each mold about 3/4 full. Don't overfill, as the madeleines will rise during baking.
Bake for 11-13 minutes, or until the edges are golden brown and the madeleines spring back lightly when touched. Keep an eye on them, as baking times can vary depending on your oven.
Remove the madeleines from the oven and let them cool in the pan for a few minutes before gently inverting them onto a wire rack to cool completely. Serve warm or at room temperature.