1. In a bowl, whisk the 1 1/2 cups of pastry flour with the 2 tablespoons of granulated sugar. Add the butter and, using a pastry blender, your fingers or 2 dinner knives, blend the butter into the flour until the mixture resembles coarse meal. Sprinkle on the vanilla and knead gently until a dough forms. Pat the dough into a disk, wrap in plastic and refrigerate until thoroughly chilled, at least 1 hour.
2. Preheat the oven to 350°. Line 2 large baking sheets with parchment paper and refrigerate. Cut the dough into 4 pieces. Working with 1 piece at a time and leaving the rest in the refrigerator, roll out the dough 1/8 inch thick on a lightly floured work surface. Using a 4-inch round biscuit cutter, stamp out as many rounds as possible; gather and reserve the scraps.
3. Brush the edges of each round with some of the egg white. Spoon 1 rounded teaspoon of jam onto one half of each round, leaving a 1/4-inch border; fold the other half over to enclose the jam. Using the tines of a fork, press the edges together to seal. Transfer the mezzalune to one of the chilled baking sheets, spacing them 2 inches apart. Repeat with the remaining 3 pieces of dough and filling, then repeat with the gathered scraps. Refrigerate the mezzalune for at least 10 minutes.
4. Brush the mezzalune with egg white, sprinkle with granulated sugar and top with a generous sprinkling of sliced almonds. Bake in the upper and lower thirds of the oven for about 40 minutes, until golden brown, switching the pans halfway through for even baking. Transfer the cookies to a rack to cool. Dust with confectioners sugar and serve.