Melt 1/2 cup + 1 tbsp of unsalted butter and set it aside to cool slightly.
Add 2/3 cup + 2 tbsp of granulated sugar, 2 room temperature eggs and 1 egg yolk to a large bowl. Beat the eggs and sugar on high with an electric mixer for 5-8 minutes, or until the mixture is pale yellow and shiny. This step helps to ensure the tops of the madeleines dome properly.
Next add in 1 tsp of vanilla and 1 1/2 tsp of lavender paste and mix just until combined.
Cut open 3 Earl Grey tea bags and add the tea leaves to a food processor. Grind the tea to ensure no large leaves are left.
In a small bowl sift together 1 cup + 2 tbsp of all purpose flour, 3/4 tsp of baking powder, 1/4 tsp of salt and the tea leaves.
Slowly add the dry ingredients to the egg/sugar mixture and fold in until everything is combined.
Next pour in the cooled but still liquid butter and 2 tbsp heavy cream and fold in just until combined. Don't over mix the batter.
Cover the bowl with plastic wrap and chill in the fridge for 1 hour.
Preheat the oven to 350 degrees. Transfer the batter to a piping bag. Brush melted butter into each madeleine mold. Pipe about 1- 1 1/2 tbsp of batter into each mold.
Bake the madeleines for 12-14 minutes until they have domed.
Remove them from the oven and let them cool in their pan for a few minutes, then transfer them to a cooling rack to completely cool.
For the Icing
While they cool make the icing by combining 2 cups of sifted powdered sugar and 3-4 tbsp of water. The icing should be thick, but if it is still really dry with the 4 tbsp of water, add a little more a tsp at a time.
Then add in 1 tsp of lavender paste and a few drops of purple food coloring to your preference.
After they have cooled, dip each madeleine into the icing and transfer to parchment paper to let the icing set and harden. Once the icing has hardened, serve the madeleines.
Store left overs in an air tight container, but unfortunately they will only be best on day 1 of baking.