Preheat oven to 350F.

In a small saucepan, melt 2 tablespoons butter and divide between two 1-cup ramekins. Sprinkle brown sugar and walnuts over butter. Cut banana into 1/4-inch-thick slices and arrange over walnuts, overlapping and pressing lightly to fit.

In a small bowl, whisk together flour, baking powder, cardamom, and a pinch salt. In a bowl with an electric mixer, beat together remaining tablespoon butter and granulated sugar until combined well. Beat in egg and vanilla until combined and beat in flour mixture until batter is just combined. Divide batter between ramekins and bake on a baking sheet in middle of oven until a tester comes out clean, about 25 minutes. Run a sharp knife around edges of ramekins and invert cakes onto plates.

Serves 2.