In a large bowl, using an electric mixer, cream butter and sugar until light and fluffy. Add the egg and vanilla.
In a separate bowl, whisk together flour, baking powder, and salt. Alternately, add the flour mixture and milk to the butter-sugar mixture. Beat until well incorporated. Scrape the bowl as needed. Form the dough into a ball and wrap with plastic paper. Refrigerate for at least one hour, or overnight.
In a medium saucepan, combine all the ingredients for the filling. Bring the mixture to a boil, stir constantly. Reduce the heat and simmer for 10 minutes, occasionally stirring. Remove from heat and allow cooling.
Preheat oven to 350°F. Line two baking sheets with parchment paper.
Remove the dough from the refrigerator. Cut the dough in quarters. On a lightly floured surface, roll each of the quarters into 12x8-inch rectangles. Cut each rectangle into 3x4-inch pieces. Place 8 rectangles in each of the prepared baking sheets. Spread 2 tablespoons of apple filling in the center of each piece. Brush the edge of the rectangles with water. Place the rest of the rectangles on top of each tart and seal de edges all around using a fork. Prick the top of each tart to allow steam to escape.
Bake for 15 minutes or until golden brown. Remove from oven and transfer to a cooling rack.
Prepare the icing by mixing powdered sugar with milk until smooth. Drizzle icing over cooled tarts.