Preheat oven to 350 F. Bake cardamom pods on a rimmed baking sheet until fragrant, about 10 minutes. Transfer sheet to a wire rack; let cool completely. Finely grind cardamom in a spice grinder. Pass through a large-mesh sieve; set aside. Make streusel: Whisk flour, brown sugar, ¾ tsp ground cardamom, and the salt in a medium bowl. Cut in butter with a pastry blender until mixture resembles crumbs; set aside. Coat a 10-inch nonstick angel-food-cake pan with cooking spray; set aside.
Make the cake: Sift flour, baking powder, baking soda, ¾ tsp ground cardamom, the cinnamon, and salt into a medium bowl. Put butter and sugar into the bowl of an electric mixer fitted with the paddle attachment; mix on medium-high speed until pale and fluffy. Reduce speed to medium. Mix in eggs, 1 at a time. Reduce speed to low. Add flour mixture in 2 batches, alternating with the yogurt. Mix in vanilla. Spoon half of the batter into prepared pan. Sprinkle with 2 ½ cups streusel. Top with remaining batter. Squeeze remaining streusel into large and small clumps; sprinkle on top of batter. Bake until golden brown and a cake tester inserted into center comes out clean, 50 to 60 minutes. Transfer to a wire rack; let cool 10 to 15 minutes. Remove cake from pan by slightly raising removable center piece; invert onto baking sheet, and then reinvert onto wire rack to cool completely.