MAKING THE BABA DOUGHDilute the yeast in three-quarters of the egg in the bowl of the stand mixer and add the flour and sugar. Mix at low speed to make a smooth dough. Increase the speed to medium for 5 minutes and add the remaining egg. Continue to mix until the dough begins to pull away from the sides of the bowl and reaches a temperature of 77°F (25°C). Add the butter and salt and mix at medium speed until the dough pulls away from the bowl again and slaps against the sides (79°F/26°C). Grease the mold with the non-stick spray. Shape 9 oz. (250 g) of dough by hand, making a hole in the center, and place it in the mold. Tap the mold on a work surface a few times to remove any air bubbles. Cover, and let rise at 90°F (32°C) for about 45 minutes. Preheat the oven on convection setting to 325°F (170°C/Gas mark 3) and bake for 20 minutes. Unmold and put the baba back in the oven for 5 minutes. Remove and let it dry out for 2 days at room temperature. Store in an airtight container.
MAKING THE ROSE-FLAVORED MASCARPONE CREAM
A day ahead, soak the gelatin in the water for at least 20 minutes. Whisk the egg yolk with the sugar until pale and thick. Bring the cream to a boil and gradually whisk it into the yolk mixture. Pour back into the pan and heat to 185°F (85°C), as for making a custard. Squeeze excess water from the gelatin and add, along with the mascarpone, rose syrup, and rose extract. Blend until smooth. Press a piece of plastic wrap over the surface of the cream and keep in an airtight container in the refrigerator overnight.
MAKING THE RASPBERRY AND ROSE BABA STEEPING SYRUP
Bring the water, sugar, and raspberry purée to a boil. Add the rose syrup, extract, and raspberry eau de vie. Remove from the heat and allow to cool. Use at 122°F (50°C) or set aside in the refrigerator.
STEEPING THE BABA
In a large saucepan, heat the steeping syrup to 122°F (50°C). Immerse the baba in the syrup, turning it over and basting with syrup from time to time. Once the baba is thoroughly soaked, lift it from the pan with a skimmer and place on a wire rack over a baking sheet. Baste it generously with the raspberry eau de vie. Let drain and chill for 2 hours.
ASSEMBLING THE BABA
Drain the lychee pieces thoroughly in a fine-mesh sieve. In the bowl of the stand mixer fitted with the whisk, beat the rose mascarpone cream until stiff. Stick the 2 plates together using glucose from the decorating bag. Baste the baba generously with the raspberry eau de vie and chill. Brush the chilled baba with warm glaze, ensuring it is not too hot or the glaze will not be absorbed. Divide the mascarpone cream between the 2 pastry bags. Place the baba on the gold plates and using the pastry bag with the plain tip, pipe mascarpone cream halfway up the inside of the cake. Sprinkle the chopped lychees and whole fresh raspberries generously over the cream. Cover the fruit with cream, creating a slight dome shape. Using the bag with the Saint-Honoré tip, pipe flame shapes on the baba in a circular pattern, working from the outside toward the center of the cake. Arrange the rose petals attractively around the piped cream, and place a fresh raspberry in the center.