1. Place 1 stick of the butter in a medium saucepan over medium-low heat. Stir occasionally with a rubber spatula as the butter melts and continue to cook until it gets golden brown, 5 to 7 minutes. Pour the browned butter into a medium bowl and let cool slightly.
2. In a small bowl, whisk together the all-purpose flour, baking powder, and salt until thoroughly combined. Set aside.
3. In a clean and dry bowl of a stand mixer fitted with the whisk attachment, whip the eggs and granulated sugar on medium-high speed until the mixture reaches the ribbon stage, 8 to 10 minutes. This results in a thick and pale yellow mixture. Whisk in the lychee extract.
4. Using a large sieve, sift one-third of the dry ingredients into the bowl with the egg mixture. Gently fold them into the egg mixture using a rubber spatula. Repeat the process twice more until all of the flour mixture is incorporated into the batter.
5. Measure ½ cup of the batter and add it to the medium bowl with the cooled butter. Proceed to whisk it into the butter until it becomes a homogenous mixture. Fold the tempered butter mixture into the remaining batter until well combined. Cover the bowl with plastic wrap and refrigerate overnight. Chilling the batter overnight is key for creating those classic madeleine humps.
6. Position a rack in the middle of the oven and preheat the oven to 350°F.
7. Place the remaining 2 tablespoons butter in a small microwave-safe bowl and microwave in 10-second intervals until completely melted. Using a pastry brush, brush the wells of a madeleine pan with the melted butter. Drop 1 heaping tablespoon of batter into each well. Bake for 10 to 12 minutes, or until the edges are golden brown. Immediately remove the madeleines from the pan and place shell-side up on a wire rack to cool completely. Repeat the process until all the madeleine batter is baked.
8. In a small saucepan, heat the lychee juice over medium-high heat. Once it reaches a boil, add the hibiscus tea bag and turn off the heat. Let the tea bag sit for at least 10 minutes. Discard the tea bag and pour the steeped juice in a small bowl.
9. In another small bowl, whisk together the powdered sugar and 3 tablespoons of the lychee juice, adding up to 1 more tablespoon of the lychee juice until it becomes a smooth glaze.
10. Take each madeleine and dip at a diagonal into the glaze. Put madeleines back on the wire rack and immediately sprinkle each with dried rose petals. Leave on the rack for at least 30 minutes, until the glaze has set. Serve at room temperature. Store any leftovers in an airtight container at room temperature for up to 3 days.