For a softer dough, add ¼ cup of all-purpose flour and skip the cornmeal.
Make the filling: Preheat oven to 350 degrees. In a medium saucepan over medium heat, combine plums, sugar, and salt. Heat until sugar and salt are dissolved and juices from plums are nearly boiling. Remove from heat and set aside.
Make the batter: In a large bowl, whisk together flour, cornmeal, sugar, baking powder, salt, cinnamon, and cardamom. Slowly whisk in buttermilk until just combined.
Assemble the cobbler: Scatter butter in a large cast-iron skillet or 9x13-inch baking dish. Place pan in oven to melt the butter, then carefully pour in batter and smooth into an even layer. Spread plums and their juices over batter. Bake until batter is golden brown and has puffed up over plums, about 40 minutes. Remove and cool cobbler on a wire rack for 30 minutes. Serve with ice cream.