Preheat oven to 350°F and butter a 9-inch pie plate or cake pan.
Mix together 1 tablespoon sugar, cinnamon, and cardamom and sprinkle over the sliced apples. Toss together until evenly coated, then spread the apples into the bottom of the pan.
In a large bowl, stir together the melted butter and sugar until smooth. Beat in the egg and vanilla.
In another bowl, sift together flour, salt, and baking powder, then stir into butter mixture. Batter will be somewhat thick.
Spoon batter over apples and spread as best you can to cover all areas.