Preheat oven to 350°F. Grease and flour a 9-inch springform pan and set aside.

To make the cake, cream together butter and 3/4 cup sugar until light and fluffy, then beat in eggs one at a time. Beat in vanilla and yogurt.

In another bowl, mix together 1 1/2 cup all purpose flour, 2 tablespoons instant espresso, baking powder, baking soda, 1 teaspoon cinnamon, 1/2 teaspoon cardamom, and salt. Mix into the wet ingredients until fully incorporated. The batter will be a bit thicker than standard cake batter.

In a small bowl, stir together the brown sugar, 1 teaspoon instant espresso, 1/2 teaspoon cinnamon, and 1/8 teaspoon cardamom.

In another bowl, mix together 1/2 cup all purpose flour, 1/2 cup sugar, and 1 tablespoon instant espresso. Mix in the melted butter with a fork until blended and clumps form. Use the tines of the fork to make the mixture quite clumpy.

Spoon half the cake batter in the prepared pan. Spoon the brown sugar mixture over the top, then add the remaining cake batter. Scatter the crumble over the top of the cake.

Bake for about 50 minutes, until cake is pulling away from the edges of the pan and a toothpick inserted in the center comes out clean.