Preheat the oven to 180°C (350°F).
Beat the softened butter and granulated sugar in a large mixing bowl until creamy. Add the eggs, crème fraîche, all-purpose flour, almond flour, almond extract, and vanilla extract. Beat again until smooth and evenly combined.
Line a 28 cm (11-inch) tart pan with the pastry crust. Press it gently against the bottom and sides of the pan, then prick the base several times with a fork.
Spread the almond filling evenly over the unbaked pastry crust.
Arrange the drained and dried apricot halves over the almond filling, pressing them gently into the surface.
Bake for about 25 minutes, or until the almond filling is set and lightly golden. Begin checking the tart after 15 minutes. If it browns too quickly, loosely cover the top with aluminum foil for the remainder of the baking time.
Once cooled, dust with powdered sugar. If using sliced almonds, toast them briefly in a dry skillet until lightly golden, then sprinkle them over the tart before serving.