Sift the bread flour, cake flour, salt and baking soda together into a medium bowl.
Put the butter, brown sugar, granulated sugar and vanilla paste together in a large bowl with an electric mixer and beat on medium-high speed until light and fluffy, about 1 minute. Add the eggs one at a time, beating well after each addition. Add the flour mixture in 2 batches and mix on medium until fully incorporated. Then add the white chocolate chunks and peppermints. Refrigerate the dough for 24 hours before baking.
Adjust the oven racks to the upper and lower thirds of the oven and preheat to 400 degrees F. Line 2 cookie sheets with parchment.
Drop heaping tablespoonfuls of the chilled batter onto the parchment-lined cookie sheets with about 1 1/2 inches of space between them. Bake until the bottoms and edges are golden brown, 10 to 12 minutes. Sprinkle the warm cookies with the remaining ground peppermints and allow to cool on a wire rack.