Cut 1/2 cup of the butter into 1/2-inch pieces and refrigerate it along with the shortening. In the bowl of a food processor, fitted with the steel blade, pulse together the cake flour, 3/4 cup of the all-purpose flour, baking powder, baking soda and 1 teaspoon salt. Add the butter and shortening and pulse to combine, until small crumbles are present. 6 to 8 times, or Add the buttermilk and pulse until a dough ball forms, about 5 to 6 times.
Dust a dry work surface with flour and have the remaining 1 cup ready for kneading. Turn the dough out onto the floured work surface. With floured hands, gently press the dough out into an 11 by 8-inch rectangle, about 1/4 to 1/2-inch thick, dust lightly with flour and gently fold it over in thirds, like a letter. Roll or press out to a 6 by 6-inch rectangle again about 2 inches thick. Cut the dough into 9 squares (2 by 2 inches each) with a pizza cutter. Transfer the biscuits to a silicone mat or parchment paper lined baking sheet about 1/2-inch apart. Cover loosely with plastic wrap and let them rest at room temperature for 15 minutes.
Preheat oven to 500 degrees F. Melt the remaining tablespoon butter and brush tops with melted butter, then sprinkle with the remaining 1/4 teaspoon of sea salt.
Place baking sheet in the middle of the oven and immediately turn oven down to 450 degrees F. Bake for 12 to 16 minutes until they are golden brown. Remove biscuits to a wire rack to cool for a few minutes.