To make the pastry, tip the flour, butter and a pinch of salt into a food processor. Pulse until the mixture resembles breadcrumbs. Add the sugar and pulse again. Add the vanilla, egg and 1-2 tbsp ice-cold water, and pulse until the dough just comes together. Tip out and shape into a disc. Roll out the pastry on a lightly floured surface and line a 22cm loose-bottomed tart tin, pressing it into the sides. Chill for 30 mins.
Meanwhile, prepare the frangipane. Using an electric whisk, beat the butter until creamy, then add the sugar and continue beating until light and fluffy. Gradually add the eggs, beating well after each addition, then stir in the almonds, flour, ginger and 1 tbsp ginger syrup.
Heat oven to 180C/160C fan/gas 4 and put in a baking tray to heat. Spoon the frangipane into the tart case and smooth. Poke the apricots into the frangipane. Transfer to the baking tray and bake for 40-50 mins (cover with foil after 30 mins if the tart is taking on too much colour) until the fruit is tender and a skewer poked in the frangipane comes out clean.
Leave the tart to cool in the tin for 10 mins, then transfer to a wire rack. Heat the remaining ginger syrup in a pan until thickened, and glaze the tart with it. Serve warm or cold, dusted with icing sugar, with a dollop of crème fraîche mixed with a little vanilla bean paste, if you like.