Line a baking tray with baking parchment. Roll out half of the pastry on a lightly floured work surface until it is about 25x25cm/10x10in in size and slightly thinner than a pound coin.
Cut a 23cm/9in circle from the pastry by cutting around the bottom of a 23cm/9in cake tin using a sharp knife. Transfer the pastry circle to the prepared baking tray and cover with cling film.
Repeat the process with the other half of the pastry, placing the second pastry circle on top of the cling film and covering with more cling film. Chill until needed.
For the filling, beat the softened butter until creamy using a wooden spoon or freestanding mixer. Add the sugar and beat again thoroughly until the mixture is pale and fluffy.
In a small bowl, beat the egg with the egg yolk until just combined. Gradually add it to the creamed butter and sugar mixture, a tablespoon at a time, beating well after each addition to incorporate the egg into the mixture.
When the mixture is very light in colour and texture, gently stir in the ground almonds, flaked almonds and almond extract. Cover the bowl with cling film and chill in the fridge for 15-20 minutes.
When the pastry and filling have cooled, remove the cling film from the pastry and leave one pastry circle on the baking tray. Spoon the filling mixture onto the pastry circle on the baking tray, heaping it into a slight mound in the middle and leaving a 2cm/1in border all around the edge. Brush the pastry border with some of the beaten egg, then gently lay the second pastry circle over the filling, pressing the edges firmly together to seal.
Holding a small knife blade at right angles to the pastry border, ‘knock up’ the edge of the galette by making small indentations all around the edge of the pastry. Scallop the pastry border by pushing the indentations in at 2cm/1in intervals using the back of the knife. Brush the top of the galette all over with more of the beaten egg, then chill the galette in the fridge for 30 minutes.
Preheat the oven to 220C/200C Fan/Gas 7.
Brush all over the top of the chilled galette again with the remaining beaten egg, then score it with a criss-cross pattern using the tip of a sharp knife. Make a couple of small steam holes in the centre of the galette. Bake in the oven for 25-30 minutes, or until the pastry is crisp and golden-brown. Set aside to cool slightly before serving.