Prepare the Pastry
In a large bowl, whisk together the flour, ground almonds, powdered sugar, and salt.
Add the cold cubed 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 enough cold water to bring the dough together. Shape into a disc without kneading.
Wrap the pastry and refrigerate for 1 hour.
Roll the chilled pastry on a lightly floured surface into a circle large enough to line a 25 cm (10-inch) tart pan. Press it into the pan and trim the edges.
Prick the base 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, then remove the weights and bake for another 10 minutes, until lightly golden. Set aside.
Make 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 and creamy.
Assemble the Tart
Spread the apricot jam evenly over the baked pastry shell.
Spoon or pipe the frangipane mixture over the jam and smooth the top.
Arrange the drained apricot halves 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 the tart in the pan for about 30 minutes, then carefully transfer it to a wire rack.
Prepare the Glaze
Heat the apricot jam and water 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.