Preheat oven to 400°. Line a baking sheet with parchment paper or a silicone baking mat.
In a food processor (or in a bowl with and electric mixer) combine almond paste, butter and sugar until crumbly. In a small bowl beat the egg until combined. Add half of the egg and the flour to the almond paste mixture and reserve the other half of the egg for brushing on the pastry.
Roll out the puff pastry to a 16" by 16" square. Cut in half, then cut along the other side to make 8 rectangles.
Place a scoop of almond filling on the lower half of each rectangle, then spread almost to the edges. Fold over the top half and press the sides together firmly.
Make four small slits along the long side to make 5 little claws. Move to the baking sheet.
Brush each claw with the remaining egg. Sprinkle with sliced almonds and a pinch of coarse sugar.
Bake at 400°F until golden, about 15 minutes. Remove from oven and transfer to a cooling rack. Some of the filling will probably ooze out of the claws, just cut it off with a knife and nibble on it (baker's privilege).
Optional Glaze: If desired, while pastries are baking combine powdered sugar with enough almond extract to make a glaze. Drizzle over the bear claws after they have baked and cooled for at least 5 minutes.