Preheat the oven to 400 degrees F. Butter the bottom and sides of an 8-inch square baking pan with some of the butter, then dust with flour and line with parchment paper.
Combine the flour, cornmeal, baking soda and salt in a large bowl. Whisk together the butter, buttermilk, cream, vinegar and eggs in a medium bowl until smooth, then quickly pour the wet ingredients over the dry ingredients and whisk to combine. Quickly pour the batter into the prepared pan, smooth the top and bake until light brown on top, firm to the touch and a toothpick inserted in the middle comes out clean, about 30 minutes.
Transfer the baking pan to a rack and brush the top lightly with more buttermilk. Let cool for 5 minutes, then invert the cornbread onto a cutting board and remove and discard the parchment paper. Flip the cornbread back over again, cut into large squares and serve while warm.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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