Preheat your oven to 400°F (200°C) and line a baking sheet with parchment paper.
On a lightly floured surface, roll out the 1 sheet puff pastry to smooth any creases and to slightly enlarge it.
1 sheet puff pastry
Cut the puff pastry into 6 equal rectangles and place them on the prepared baking sheet.
Using a knife, lightly score a border about 1/2 inch from the edge of each rectangle, being careful not to cut all the way through. This will help the edges puff up.
Brush the edges of the pastry with the beaten 1 egg.
1 egg
Bake in the preheated oven for 12-15 minutes, or until the pastry is golden brown and puffed. Once baked, allow them to cool completely on a wire rack.
In a blender or food processor, puree the diced 2 ripe mangoes until smooth.
2 ripe mangoes
In a medium saucepan, combine the mango puree, 1/2 cup granulated sugar, 3 egg yolks, and 1/4 cup fresh lime juice. Cook over medium heat, stirring constantly, until the mixture thickens enough to coat the back of a spoon, about 8-10 minutes.
1/2 cup granulated sugar, 3 egg yolks, 1/4 cup fresh lime juice
Remove from heat and stir in the 4 tbsp unsalted butter, one piece at a time, until fully incorporated.
4 tbsp unsalted butter
Transfer the mango curd to a bowl, cover with plastic wrap (pressing the wrap directly onto the surface to prevent a skin from forming), and refrigerate until fully cooled and thickened.
In a small bowl, combine the 1 cup fresh raspberries, 1 tbsp granulated sugar, and 1 tsp lemon juice.
1 cup fresh raspberries, 1 tbsp granulated sugar, 1 tsp lemon juice
Gently stir to coat the raspberries evenly, then let them sit at room temperature for about 15-20 minutes to release their juices and soften.
In a small bowl, whisk together the 1/2 cup powdered sugar, 1 tbsp milk, and 1/2 tsp vanilla extract until smooth. The icing should be thick yet pourable; add a bit more milk if needed.
1/2 cup powdered sugar, 1 tbsp milk, 1/2 tsp vanilla extract
Once the puff pastry rectangles have cooled, gently press down the center of each tart with the back of a spoon to create a shallow well.
Spoon the chilled mango curd into the well of each tart.
Top each tart with a spoonful of macerated raspberries, allowing some of the juices to drizzle over the curd.
Using a spoon or a small piping bag, drizzle the vanilla icing over the assembled tarts.