Crust:
Finely chop the nuts in a food processor in pulse mode, but so that some large pieces are still visible.
Transfer the nuts to another container and reserve.

Put the flour, powdered sugar, and salt in the food processor and turn it on to mix well.
Add the butter until the mixture resembles coarse breadcrumbs.
Add 2 tablespoons (30 ml) of rum and vanilla, and turn it on briefly to incorporate. Add the nuts and turn the processor on again in pulse mode, just enough to distribute them evenly. If the dough is too dry, add the last tablespoon (15 ml) of rum.

Form the dough into a ball, then into a disk, and wrap it in plastic wrap. Let it sit at room temperature while preparing the mangos.

Mangos:
Peel the mangos with a vegetable peeler and cut the pulp into long slices 3/4 inch (2 cm) thick.
Mix the mango slices with lime juice.

Melt the butter in a large skillet of 9 to 10 inches (23 to 25 cm) that can go in the oven, over medium heat.
Sprinkle the skillet with sugar, place the mango slices, and drizzle with rum.
Level the mango slices in the skillet with a spatula.
Cook without stirring for 10 to 12 minutes.

Meanwhile, preheat the oven to 400 F (200 C).
Roll out the dough on a lightly floured surface to obtain a circle of 10 inches (25 cm).

Cover the hot mangos with the rolled-out dough, folding the edges towards the inside of the skillet.
Cook in the middle of the oven for 25 minutes, until the crust is golden brown.

Remove the tart from the oven. Let it cool for 5 minutes to make the crust crispy.
Loosen the edges with the tip of a knife.
Place a plate upside down on the crust in the skillet.
Flip the tart onto the plate and replace the mango pieces that stayed stuck to the bottom of the skillet. Let it cool without covering.

This tart is best served the same day it is made.