Combine the flour, sugar, lime zest and salt in a food processor and pulse several times to combine. Add the chilled and grated or cut pieces of butter to the food processor and pulse until the mixture resembles a coarse meal.With the processor running on low, add 1 tbsp of Rum at a time until the mixture comes together into a dough ball. Turn it out onto a a lightly floured surface and shape it into a flattened disc with your hands. Cover it with plastic wrap and refrigerate for 3 hours or overnight.Preheat oven to 400 degrees F when ready to make the filling, and line a baking sheet with parchment paperIn a large bowl combine all the filling Ingredients and gently stir to combine.Sprinkle flour on a your rolling surface along with your rolling pin and get the disc dough out of the fridge and roll it out between two pieces of parchment paper into approx 12 inch round. Once you have it rolled out transfer it onto a baking sheet then peel off the top layer of parchment paper.Sprinkle 1 tsp of flour evenly over the middle of the pie dough. Starting with the ripest mangos, spoon them out into the middle of the dough and the more firm pieces you can arrange on top, and spread out leaving 1½ inch border all around.Fold the edges of the dough as if you are pleating it as you work your way around the whole galette which will hold all that lovely filling in.Whisk the egg and brush the top of the dough with it and then sprinkle the dough generously with the brown sugarBake for approx 30-35 minutes or until the filling is bubbling and the crust is golden brown. It’s totally ok if juice from the filling seeps out a bit onto the parchment paper while it’s baking. Serve warm topped with ice cream or whipped cream or enjoy it on its own.