Getting ready: Center a rack in the oven and preheat the oven to 350 degrees F. Butter a 3-inch-high 8-inch springform pan, dust the inside with flour, tap out the excess and line the bottom with parchment paper. Put the pan on a baking sheet lined with parchment paper or a silicone mat.
To make the cake: Sift together the flour, baking powder and salt.
Place the eggs and sugar in the bowl of a mixer of another large heatproof bowl and whisk to blend. Put the bowl in a skillet with an inch or two of simmering water and continue to whisk until the sugar dissolves completely and the mixture is just warm to the touch, about 3 minutes. Remove the bowl from the heat and whisk in the vanilla.
Working with a stand mixer fitted with the whisk attachment or with a hand mixer, beat on medium speed for about 5 minutes, until the mixture almost triples in volume and forms a ribbon that holds it shape for about 10 seconds when the beater is lifted.
Switch to a large rubber spatula, sift over half the dry ingredients and fold them in gently- don't overmix, you'll have another chance to gather up any recalcitrant ingredients. Fold in the cooled, melted butter, then sift over the remaining dry ingredients and very gingerly fold them in. The beautifully beaten eggs are fragile and must be treated like the divas they are. Check that you don't have a puddle of melted butter at the bottom of the bowl-a frequent hazard- then scrape the batter into the pan and jiggle the pan gently to even it.
Bake the cake for 30-33 minutes, or until the top is springy to the touch and the sides are starting to pull away from the pan. (A thin knife inserted into the center would come out clean, but it would also deflate the still-fragile cake a little.)
Transfer the cake to a rack and let it sit for 5 minutes, then run a blunt knife between the cake and the pan and remove the sides of the pan. Invert the cake and remove the base of the pan and peel off the paper, then invert again and cool to room temperature right side up. ( When cool, the cake can be wrapped and kept at room temperature overnight- it will cut a little more easily if it rests- or wrapped airtight and frozen for up to 2 months)
To make the syrup: Stir the water and sugar together in a small saucepan, bring to a boil and boil for 1 minute. Pour the syrup into a heatproof bowl. Stir in the liqueur and let the syrup come to room temperature.
To make the filling: Working with a stand mixer, preferably fit with a paddle attachment, or with a hand mixer in a large bowl, beat the cream cheese at medium speed until soft, smooth and fluffy. While beating, gradually add 1/2 cup of the cream, the sugar and vanilla and continue to beat until the cream is absorbed and the cheese is smooth. Scrape the mixture into a medium bowl. Pour the remaining 2/3 cup cream into the bowl you beat the cream cheese in (there's no need to wash it) and, using the whisk attachment or the hand mixer, whip the cream until it holds firm peaks. Stir about one quarter of the whipped cream into the cream cheese mixture, then fold in the rest. There's no need to wash the bowl-you'll be using it for the topping.
To make the topping: Just before you are ready to assemble the cake, whip the cream until it holds medium peaks. Add the sugar and vanilla and continue to whip until the cream holds firm peaks. Cover the cream and refrigerate.
To assemble the cake: Using a serrated knife, slice off the top 1/2 inch of the cake and set this layer aside. If it crumbles- that can happen to the best of us- just save the pieces. With the knife, sketch a circle on top of the cake that is 1/2 inch in from the edges of the cake. Cut down through the outline, stopping between 1/4 and 1/2 inch from the bottom. Using the knife and your fingers, carefully pull out the cake within this circle, leaving a little nest that will house your surprise. Transfer the cake nest to a platter or a cardboard round.
Brush the inside of the cake with some of the syrup, discarding what remains, and spoon a thin layer of the filling over the bottom. Toss in 1/2 pint of the berries-more if you'd like- and cover with the remaining filling. Lift the reserved top layer onto the cake and press it down gently to form the cake. If all you have are scraps, fret not; just arrange them over the top of the cake and press them gently to reconstruct a layer.
Using a long metal icing spatula, frost the top and sides of the cake with the topping. Finish with some raspberries, placing berries carefully around the top of the cake, or piling them up in the center. Refrigerate the cake for at least 1 hour (or for up to 6 hours) before serving.