Prepare the Pastry
In a large bowl, whisk together the flour, ground almonds, powdered sugar, and salt.
Add the cold vegan butter and rub it into the flour mixture with your fingertips until it resembles fine breadcrumbs.
Stir in the cold vodka (if using), then gradually add just enough cold water until the dough comes together. Shape into a disc without kneading.
Wrap the dough and refrigerate for 1 hour.
Roll the chilled pastry into a circle large enough to line a 25 cm (10-inch) tart pan. Press the pastry into the pan and trim the excess.
Prick the base all over with a fork and freeze for 20 minutes.
Preheat the oven to 350°F (180°C) or 325°F (160°C fan).
Line the pastry with parchment paper or foil and fill with pie weights or dried beans.
Blind bake for 20 minutes. Remove the weights and bake for another 10 minutes until lightly golden. Set aside.
Prepare the Frangipane
In a large bowl, whisk together the ground almonds, flour, cornstarch, baking powder, and sugar.
Add the melted vegan butter, non-dairy milk, and almond extract. Whisk until smooth.
Assemble the Tart
Spread the apricot jam evenly over the cooled tart shell.
Spoon or pipe the frangipane mixture over the jam and smooth the surface.
Pat the apricot halves dry with paper towels and arrange them evenly over the frangipane.
Sprinkle the sliced almonds over the top.
Bake for 1 hour, or until the frangipane is golden and a skewer inserted into the center comes out clean.
Cool in the pan for about 30 minutes, then carefully remove from the tart pan and transfer to a wire rack.
Make the Glaze
Heat the apricot jam and water together in a small saucepan until melted.
Strain through a fine-mesh sieve for a smooth glaze.
Brush the warm glaze over the cooled tart and allow it to set before serving.