Preheat oven to 400°F.
Whisk buttermilk, egg and 3 tablespoons melted butter in a medium bowl.
Place flour, baking powder, baking soda and salt in a stand mixer fitted with a paddle attachment. Mix on low speed to combine. Increase speed to medium and slowly add the buttermilk mixture until a soft dough forms. Let the dough rest for 10 minutes.
Brush the remaining 1 tablespoon butter over the base and partway up the sides of an 8-inch cast-iron skillet. Dust hands with flour and form the dough into a ball. Press the dough into the prepared pan. Use a sharp knife to score a deep "X" into the top of the dough.
Bake until lightly browned and hollow sounding when tapped on the bottom, 30 to 40 minutes. Remove from the pan and let cool on a rack for at least 30 minutes.
