For the feuille de brick tubes: Preheat a convection oven to 160°C/325°F; turn the fan off if possible.
Lightly spray the exterior of 10 metal tubes measuring 7.5cm/3in long by 2.5cm/1in diameter with nonstick oil spray. Place them on a sheet pan in a standing position.
Combine the melted butter with the vanilla paste; reserve warm.
Cut out 20 pieces of silicone paper. Ten should measure 7.5cm/3in wide by 7.5cm/3in long, and 10 should measure 7.5cm/3in wide by 10cm/4in long. Line the sprayed tubes with the smaller rectangles of silicone paper. Brush a small amount of the butter and vanilla paste mixture around each parchment-lined tube.
Using a ruler and a paring knife, cut out strips of brick dough measuring 5cm/2in wide by 7.5cm/3in long. Wrap the brick dough around the prepared tubes. The length of the dough should make it overlap itself twice and create a 5-cm/2-in long tube.
Wrap each tube with the longer sheets of silicone paper; keep the paper in place using a paper clip. Return the tubes to a standing position. If the oven fan cannot be shut off, lay the tubes down with the paper clip facing down; this will help keep the tubes from rolling around while the fan blows in the oven.
Bake until the dough is golden brown, 8 to 10 minutes with the fan on or 12 to 15 minutes with the fan off.
Once the tubes have cooled, gently remove the silicone paper from the tubes and slide the baked brick tubes from the mold. Reserve in an airtight container at room temperature. Discard after 3 days.
Place the baked brick tubes on a flat sheet pan lined with a nonstick rubber mat.
For the frozen mango parfait: Combine the heavy cream and the first amount of sugar in the bowl of an electric mixer fitted with a whip attachment. Whip to medium peaks. Reserve refrigerated.
Combine the egg yolks with the second amount of sugar in mixer bowl and cook over a hot water bath until the mixture reaches 60°C/140°F while stirring constantly with a whisk. Stir for 2 minutes at this temperature. Place the bowl on the electric mixer and whip on high speed to quadruple the volume of the yolks (ribbon stage). Meanwhile, combine the mango purée with the bloomed gelatin in a bowl and melt over a warm water bath. Do not let this mixture get too hot.
Fold the egg yolk mixture (pâte à bombe) into the mango and gelatin mixture.
Fold a small amount of the whipped cream into the egg-mango mixture, then fold in half of the whipped cream. Once this amount has been folded in completely, fold in the remaining whipped cream.
Transfer to a piping bag. At least 2 hours and up to 8 hours before service, pipe the mango parfait into the baked brick tubes. Even out the top with an offset spatula and freeze. Once frozen, cover them well with plastic wrap and reserve frozen at -10°C/14°F.