Make the dough. Pulse the flour, black cocoa, and salt in a food processor until evenly combined. Add the cold butter and pulse until the biggest pieces are pea-sized. Drizzle in 5 tbsp ice water and pulse just until the dough clumps when pinched, adding more water a tablespoon at a time if it’s dry. Cocoa soaks up water, so it will likely need more than plain pastry. Press into a disk, wrap, and chill at least 1 hour.
Cook the potatoes while the dough chills. Heat the oven to 400°F. Prick the orange sweet potato and roast it on a baking sheet for 45 to 50 minutes, until completely soft. Meanwhile, peel and cube the purple sweet potato and steam it 15 to 20 minutes, until a fork slides through. Roasting the orange one deepens the flavor. Steaming the purple one protects the color. Mash each separately, measure 1 ¼ cups of each, and let them cool.
Roll and shape the crust. Roll the dough between two sheets of parchment into a 13-inch round. Peel off the top sheet, flip the dough into the tart pan, and peel off the second sheet. Press the dough into the corners and up the sides without stretching it, patch any tears with scraps, and trim the excess. Prick the bottom with a fork and freeze 30 minutes. Set the oven to 375°F.
Blind bake. Line the cold crust with parchment and fill it with pie weights. Bake 20 minutes, then remove the parchment and weights and bake 8 to 10 minutes more. You can’t judge doneness by color here, so look for a bottom that looks dry and matte instead of shiny. Brush the hot crust with the beaten egg white and return it to the oven for 2 minutes to seal it, which keeps the bottom from going soggy. Lower the oven to 325°F.
Make the orange layer. Blend or whisk the orange sweet potato, evaporated milk, maple syrup, egg, yolk, melted butter, all the spices, salt, and vanilla until completely smooth. Pour it into the warm crust and smooth the top.
Bake. Bake at 325°F for 35 to 45 minutes, until the edges are set and the center has only a slight wobble. Since you can’t see the crust edges browning, cover them with strips of foil about halfway through. Cool on a rack 1 hour, then refrigerate 30 minutes so the surface is cold before the ganache goes on.
Make the ganache. Heat the cream until steaming, pour it over the chocolate and salt, and let it sit 2 minutes. Stir until smooth, then cool about 10 minutes, until lukewarm but still pourable. Pour it over the chilled orange layer and tilt the pan to spread it edge to edge. Chill 20 minutes, until set.
Start the purple custard. In a bowl, whisk the maple syrup, egg, cornstarch, cardamom, and salt until smooth. Heat the milk and cream in a saucepan until steaming. Slowly whisk half of the hot milk into the egg mixture, then pour it all back into the pan.
Cook it until it means it. Whisk constantly over medium heat until the custard thickens and bubbles, then keep whisking a full minute more so it sets properly in the fridge.
Blend. Add the hot custard, purple sweet potato, butter, and vanilla to the blender and blend until silky, venting the lid. Blend in the lemon zest and 1 tsp lemon juice. Taste it, and add the last ½ tsp of juice if the color looks dull or it tastes flat.
Cool it slightly, then layer. Let the custard sit 10 to 15 minutes, stirring now and then, until warm but no longer hot. Pour it slowly over the back of a spoon onto the set ganache and smooth the top.
Chill. Press plastic wrap directly onto the surface and refrigerate at least 4 hours, or overnight, until firm enough to slice.