Make the tart crust: In a bowl, whisk together the all-purpose flour, powdered sugar, and salt. Rub the cold cubed butter into the dry ingredients with your fingertips until the mixture resembles coarse breadcrumbs. Add the egg yolk and mix just until a dough forms. Shape into a disk, wrap, and chill for 30 minutes.
Roll and bake crust: Preheat oven to 350°F (180°C). On a lightly floured surface, roll the chilled dough to about 1/8 inch thick. Line the base of a round entremet ring or tart pan with the dough, pressing it in evenly. Prick the base with a fork, line with parchment, and fill with pie weights. Blind bake for 15 minutes, remove weights and parchment, and bake 3–5 minutes more until lightly golden. Cool completely in the pan.
Make the vanilla sponge: Keep oven at 350°F (180°C). In a mixing bowl, whisk the eggs and 90 g sugar with an electric mixer until pale, thick, and doubled in volume. Sift the cake flour over the mixture and gently fold it in with a spatula. Fold in the melted butter and vanilla bean paste just until combined. Spread the batter in a parchment-lined baking tray into an even thin layer. Bake for about 12 minutes until lightly golden and springy. Cool completely, then cut a round piece to fit neatly over the baked tart crust.
Make the mango passion fruit compote insert: In a small saucepan, combine 80 g mango puree, 12 g sugar, and 20 g passion fruit juice. Heat gently over low heat, stirring until the sugar dissolves and the mixture is hot but not boiling. Add the hydrated 3 g agar-agar and cook, stirring, for 1–2 minutes until fully dissolved. Pour into small round silicone molds slightly smaller than the final entremet size. Freeze until firm.
Make the mango coconut mousse: In a saucepan, combine 150 g mango puree and 50 g sugar. Heat over medium-low heat until the sugar dissolves and the mixture is hot. Add the hydrated 4 g agar-agar and cook, stirring, for 1–2 minutes until completely dissolved. Remove from heat and let cool until just warm. In a chilled bowl, whip the coconut cream to soft peaks, then whip the cold whipping cream to soft peaks in a separate bowl. Fold the whipped coconut cream and whipped cream gently into the cooled mango mixture until smooth and airy. Refrigerate until slightly thickened but still spreadable.
Make the passion fruit crémeux: In a heatproof bowl, whisk the 3 egg yolks with 90 g sugar until well combined. Heat 120 g passion fruit juice in a saucepan until hot but not boiling. Slowly pour the hot juice over the yolk mixture while whisking constantly. Return the mixture to the saucepan and cook over low heat, stirring constantly, until it thickens enough to coat the back of a spoon. Remove from heat and add the hydrated 4 g agar-agar, stirring until dissolved. Stir in the cubed butter until smooth and glossy. Transfer to a shallow dish, cover the surface with plastic wrap, and chill until set but creamy.
Make the lime white chocolate whipped ganache: Place the chopped white chocolate in a heatproof bowl. In a small saucepan, heat 100 ml heavy cream with the lime zest over medium-low heat until hot and fragrant but not boiling. Pour the hot cream through a strainer over the white chocolate, discarding the zest. Let sit 1 minute, then stir until smooth. Add the hydrated 2 g agar-agar and stir until dissolved and fully incorporated. Cover and chill until very cold and thickened, then whip with an electric mixer until light, fluffy, and pipeable. Keep chilled until ready to use.
Assemble the entremet layers: Place the cooled tart crust in an entremet ring or lined mold. Lay the cut vanilla sponge circle directly on top of the crust, pressing gently for good contact. Spread an even layer of passion fruit crémeux over the sponge and chill for 15–20 minutes until it begins to set.
Add mousse and compote insert: Spread or pipe a layer of mango coconut mousse over the set passion fruit crémeux, leaving a shallow well in the center. Unmold the frozen mango passion fruit compote inserts and press one into the center of the mousse, smoothing mousse over and around it so it is fully enclosed. Tap the mold lightly to remove air pockets, then freeze the entire entremet until very firm.
Make the passion fruit glaze: In a small saucepan, combine 100 g passion fruit juice and 50 g sugar. Heat gently until the sugar dissolves and the mixture is hot. Add the hydrated 5 g agar-agar and cook, stirring, for 1–2 minutes until completely dissolved. Pour into a bowl and let cool, stirring occasionally, until it reaches about 86°F (30°C) and is slightly thickened but still pourable.
Glaze the entremet: Remove the frozen entremet from the mold and place it on a wire rack set over a tray. Pour the cooled passion fruit glaze evenly over the top, letting it flow down to form a smooth, glossy dome. Tilt gently if needed to cover. Transfer the glazed entremet back to the freezer to set the glaze.
Decorate and serve: When the glaze is set, transfer the entremet to a serving plate. Pipe or spoon the lime white chocolate whipped ganache decoratively on top and around the edges as desired. Place the entremet in the refrigerator and thaw gently for 2–3 hours until fully defrosted but still chilled before slicing and serving.