Preheat the oven to 350°F. Line baking sheets with parchment paper and set aside.
In a medium microwave-safe bowl, combine the caramel candies and heavy cream. Microwave in 30-second increments, stirring between each interval, until smooth and creamy. Set aside to cool at room temperature.
In the bowl of a stand mixer fitted with the paddle attachment, cream the softened butter, sugar, and brown sugar on medium speed until smooth.
Add the oil, egg, and vanilla extract. Mix well, scraping down the sides of the bowl often with a rubber spatula.
Add the flour, corn starch, salt, and baking powder. Mix on low speed until thoroughly combined, scraping down the sides often.
Fold in the toffee pieces with a wooden spoon or spatula until evenly distributed.
Using a large cookie scoop, divide the dough and place scoops on the prepared baking sheets, spacing them about 3 inches apart.
Use the bottom of a glass cup or measuring cup to flatten each dough ball into a disc about ½ inch thick.
Bake for 10–12 minutes, or until the edges are just barely beginning to brown and the centers are set. Let cool completely on the baking sheets.
In another medium microwave-safe bowl, melt the chocolate chips in 30-second increments, stirring between each interval until smooth. Let cool to room temperature.
To assemble: Once the cookies are cool, spoon caramel onto each cookie and spread evenly with the back of a spoon or a small offset spatula.
Top with melted chocolate and spread evenly to cover.
Optional: Place assembled cookies in the fridge or freezer for 10–15 minutes to set the toppings faster.