In a large bowl beat the butter and sugars until fluffy.
Beat in the egg and vanilla.
With the mixer on low - beat in the flour, cornstarch, baking powder and salt - adding about ½ of the flour at a time.
Beat in the ⅔ cup toffee pieces.
Divide the dough in 2 balls.
On a piece of wax paper, form each into a log shape about 1 inch in diameter. Then roll the log in the wax paper, twist the ends and place in the fridge. Chill in the fridge for at least 4 hours or up to 4 days, or freeze.
If logs are frozen, thaw in the fridge overnight before baking.
When ready to bake, preheat the oven to 350F degrees and line cookie sheets with parchment paper or silicone baking mats.
Slice each log into discs about ¼ to ⅓ inch (about ½ centimetre) in thickness.
Place 2 inches apart on the cookie sheet and bake for 9-11 minutes, or until the tops look set.
In a small bowl melt the chocolate and shortening in short 30 second bursts in the microwave, stirring between each.
Dip half of each cookie in the melted chocolate and sprinkle with toffee bits.