Preheat the oven to 400 degrees F and butter a 9-by-13-inch baking dish. Whisk the flour, cornmeal, baking soda and salt in a large bowl.
Beat the butter in a large bowl with a mixer on medium-high speed until light and fluffy, about 2 minutes. Add the sugar and beat until well combined. Add the eggs, one at a time, beating well after each addition. Add the flour mixture in 3 batches, alternating with the buttermilk, beginning and ending with the flour mixture. Add the cheese, scallions and chives and beat until just combined.
Transfer the batter to the prepared baking dish; smooth the surface. Bake until golden brown around the edges and a toothpick inserted into the center comes out clean, about 20 minutes. Serve warm or at room temperature.