Using an electric mixer or a stand mixer (fitted with the paddle attachment), beat together the butter, white sugar, and brown sugar until light and fluffy, about 3 to 5 minutes. Scrape the sides of the bowl as needed.
Beat in the vanilla extract, almond extract, salt, and cornstarch just to combine. Add in the flour and again beat to just combine. Don't overmix.
Fold in the chopped toffee and gently knead the dough together. It will seem crumbly at first!
Divide the dough in two equal halves. Form each half into a log shape that is 9 to 9.5 inches long and about 2 inches in diameter. Roll it on the counter to help smooth it out. (Use plastic wrap to avoid touching the dough with hands and warming the butter too much.) Wrap tightly in plastic wrap and place in the fridge to chill for 2 hours.
Preheat oven to 350 degrees F. Line a baking sheet with parchment paper or a Silpat liner. Pour the cane sugar onto a large flat plate and set aside. Remove one log at a time from the refrigerator and remove the plastic wrap. Brush the outside of the dough with the beaten egg. Immediately roll the dough log into the cane sugar on the plate.
Slice each log into 1/2-inch thick cookies. Place cookies on the prepared tray. Bake until edges are *just* beginning to lightly brown, about 8-11 minutes. Watch carefully to avoid overbaked cookies!
Remove from the oven and transfer to a cooling rack. Allow to cool completely.
To drizzle in chocolate: Add chocolate to a bowl and microwave (at 50% power) in bursts of 20 seconds, stirring in between each burst for 15 seconds until the chocolate is melted and smooth.
Transfer the chocolate to a plastic bag and snip off the tip with scissors. Drizzle COMPLETELY cooled cookies with chocolate and allow the chocolate to harden at room temperature.
Cookies are best when eated within 2-3 days; store at room temperature in an airtight container.