In a small saucepan over medium heat, melt butter and mix with a rubber spatula until it has a deep brown color, about 3-5 minutes. Remove from heat immediately and transfer to a small bowl to stop cooking; set aside to cool to room temperature.
In a small bowl, combine flour, baking powder and salt; set aside. In the bowl of a stand mixer, beat eggs and sugar on medium-high speed until thickened and pale yellow in color, about 8 minutes. Add lychee extract. Using a spatula, sift flour mixture and gently fold into egg mixture in three additions.
Take about 1/2 cup of the batter and mix it into the cooled melted butter until thoroughly mixed.
Mix this tempered butter mixture into the rest of the batter, folding it in until combined. Cover with plastic wrap and refrigerate overnight.
Preheat oven to 350°F. Generously butter cavities of the madeleine pan. Scoop 1 heaping tablespoon batter into each cavity. Bake for 10-12 minutes or until the edges are golden brown.
Immediately transfer madeleines to a cooling grid, shell side up. Cool completely. Glaze each side and then top with rose petals.