Preheat the oven to 400°F and position a rack in the middle of the oven. Generously butter an 8-inch square baking dish (or any baking dish roughly that size). Set aside.
Combine the flour, cornmeal, baking powder, baking soda, cayenne, jalapeño, salt and half the grated cheddar in a large mixing bowl. Wisk well to combine. Set aside.
Combine the brown sugar, thawed corn kernels and buttermilk in the bowl of an electric food processor. Process about 5 seconds. Add the eggs and process about another 5 seconds, until well combined. (It doesn’t need to be perfectly smooth. Small bits of corn will remain.)
Using a rubber spatula, fold the wet ingredients into the dry ingredients, but give just a few turns, just to barely combine everything. Add the melted butter and continue folding until the dry ingredients are just moistened. Pour the batter into the prepared baking dish, smoothing the top surface with the rubber spatula. Sprinkle the remaining cheddar evenly over the top surface.
Bake until deeply golden brown and a toothpick inserted into the center comes out clean, about 25-30 minutes. Cool on a wire rack for 10 minutes, then invert the cornbread onto the wide rack and turn right side-up. Continue to cool 10 minutes longer, until just warm. Slice into 16 2-inch squares and serve.
