Combine the cake flour, bread flour, baking soda, baking powder and coarse salt in a large bowl. Stir to combine and set aside.
In a large mixer add the butter and sugars and mix for 5 minutes until fluffy. Add the eggs, one at a time, making sure to scrape down the sides of the bowl in between. Add the vanilla and combine.
Add the dry ingredients a little at a time and mix until just combined. Add the chocolate discs and fold them into the batter. Cover the dough and and refrigerate dough for up to 72 hours.
When ready to bake, preheat oven to 350 degrees F. Scoop 1/4 cup sized cookie dough balls onto a parchment lined baking sheet, 12 cookies per sheet and sprinkle the tops with fleur de sel. Bake the cookies for 16-20 minutes until golden brown. Remove from oven and set aside to cool.
Once cooled, scoop some ice cream and sandwich it between two cookies. Serve immediately.