Preheat your oven to 350 degrees, and line two baking sheets with parchment paper or nonstick sheets. In a medium bowl, whisk together the flour, bread flour, baking soda, and salt. Set aside. 

In a large bowl, beat together the softened butter, sugar, and brown sugar until fluffy. Beat in the eggs and vanilla until smooth. Slowly add in the flour mixture, and beat until completely incorporated. 

Scoop dough by 2 tablespoonfuls onto your baking sheet. These will spread quite a lot, so leave a good amount of space between. Bake for 8-10 minutes, or until the cookies have just started to become golden brown on the edges. Do not over bake. When you remove them from the oven, immediately sprinkle on some coarse salt. Cool completely on the cookie sheets before removing. 

Chill the cookies in the fridge once cooled to room temperature until you are ready to use them for the sandwiches. 

Line two 9x13 inch cake pans with foil. Set aside. 

In a large bowl, beat the heavy cream until stiff peaks form. You want it to be slightly over-whipped. Then, stir in the condensed milk, vanilla extract, and salt until combined. 

Moving quickly, stir in the melted butterscotch. It will immediately start to cool and harden some, so you want to stir it in quickly to ensure that it gets incorporated completely. 

Divide the ice cream between the two cake pans, and spread evenly. It should be about 3/4 inch thick, but does not have to be perfect. Freeze both pans of ice cream until solid, about 2 hours. 

Remove from the freezer, and use the foil to pull the ice cream out of the cake pan, working with only one pan at a time so it doesn't melt too quickly. Using a round cutter that is slightly smaller than your cookies, cut out rounds of ice cream, and sandwich between two cookies, pressing together slightly. 

You want to work in small batches here, and move quickly so the ice cream doesn't melt too fast! Wrap each sandwich tightly in plastic wrap or place into individual zipper bags and freeze immediately. Enjoy!