In a food processor, combine the toffee, sugar, baking soda, and salt. Blitz until the toffee is finely ground and the mixture looks like sand.
Move the toffee mixture to the bowl of your stand mixer and add the butter, egg white, and vanilla. Using the paddle attachment, mix until light and fluffy, around 2 minutes
Add the flour and mix on low until a soft dough comes together.
Divide the dough in half and pat each into a disc. Place in the fridge to chill for 15 minutes. Line three baking trays with parchment paper.
Dust your work surface with flour and place half of the dough on there. Roll out the dough, turning frequently until the dough is the ⅛" thick.
Once the dough is rolled out, use a 2" round cookie cutter, dusted in flour to cut out rounds from the sheet of dough and transfer the rounds to the prepared baking trays, leaving around ½" between the cookies.
Repeat with the other half of the dough and continue rerolling scraps of dough until all of the dough is cut into rounds. You should have around 72 rounds, if you're a little under, that's fine too.
Place the trays with the prepared rounds in the freezer for at least 30 minutes. Once firm, the cookies can be baked or stored in the fridge for use later. If storing, place parchment between the rounds and place rounds in a ziploc bag.