Cream together the butter, granulated sugar, brown sugar and salt with an electric mixer until light and fluffy, about 3 minutes.
Add each egg one at a time, being sure to fully incorporate the first one before adding the second. Add the vanilla and beat for 30 seconds.
Add the flour and baking soda and mix until just incorporated. Be sure not to overmix the dough at this point or it can get tough.
Add the reserved small bits of toffee (not the large chunks) and chocolate chunks, then mix until evenly distributed*
Preheat the oven to 350 degrees F and line a few sheet pans with parchment (without parchment the toffee may stick).
Scoop out medium-large balls (about 3-tablespoons) and place them on the sheet pan with plenty of space to spread - I fit 6 to a pan.
Place a chunk or two of toffee in the center of each ball of cookie dough and gently press in to secure.***
Bake for 7 minutes, then take the pan out of the oven and bang it on the counter, then place it back and continue cooking for 3-5 more minutes (about 10-12 minutes total), or until the cookies are golden around the outside. Bang them again once more on the counter when they're done.****
Let cool on the pan. Best eaten within 2-3 days.