Preheat the oven to 375F (190C). In a large bowl, stir together the 1/2 cup of brown sugar, the 1 tbsp of flour, ¾ tsp of the cinnamon and the cardamom. Add the apples. Toss until apples are coated.
Line a 22.5 cm (9-inch) pie plate with the pie crust. Trim crust to 1 cm (½-inch) beyond edge of pie plate. Fold under extra crust and crimp edge. Spoon the apple mixture into crust.
In a medium-size bowl, stir together the 2/3 cup brown sugar, 1/2 cup flour and the remaining 1/2 tsp cinnamon.
Cut in the butter. Sprinkle over the apple mixture. Bake for 40 to 45 minutes or until apples are tender and crumb topping is golden. Cool on a wire rack for 2 hours before serving.