Heat oven to 350 degrees. Generously butter a 9-inch springform pan. Coat the bottom of the pan evenly with light brown sugar. Toss plum slices with lemon juice, then arrange in a single layer on the sugar, skin side down. Set aside.

Put granulated sugar in the bowl of a stand mixer fitted with whisk attachment (or in a sturdy mixing bowl if using a whisk). Add the eggs and vanilla, and beat on medium until the mixture begins to thicken, about 5 minutes. Add butter. Increase speed a bit and continue beating until mixture is fluffy, another 5 minutes.

Sift together flour, baking powder, cardamom and salt. On low speed, add flour mixture to egg-sugar mixture. Stir briefly to incorporate into a smooth batter.

Spread the batter over the plums in the prepared pan. Bake until a tester inserted in the center comes out clean, about 1 hour.

Let the cake rest in the pan for 10 to 15 minutes, then invert over a cake platter. Loosen and remove pan. Cool completely to room temperature.