Prepare the caramel sauce while the cookies cool down. Microwave the caramels and heavy cream in 30-second intervals, stirring in between. Allow the caramel to cool and then chill it in the fridge while you make the cookies.
Make the cookie base by preheating the oven to 350°F. Line a baking sheet with parchment paper. In a mixing bowl, combine the softened butter, oil, brown sugar, and granulated sugar. Add the egg and vanilla, mixing well. Add the flour, cornstarch, baking powder, and salt. Mix until the flour disappears. Fold in the toffee bits.
Using a large cookie scoop, form 10 equally sized dough balls. Roll them between your palms and place them on the baking sheet, leaving about 1.5 inches of space between cookies. Gently flatten each cookie to ⅓-½ inches thickness using the back of a ¼ measuring cup. Bake in the preheated oven for 8-10 minutes. Let the cookies cool on the baking sheet for 10 minutes before transferring them to a cooling rack.
Melt the chocolate in the microwave in 30-second intervals, stirring in between.
Assemble the Twix cookies by topping each soft sugar cookie with 1-1.5 tablespoons of caramel, spreading gently. Then, spread 2-3 teaspoons of melted chocolate on top. Chill the cookies in the fridge for a few minutes before indulging in this homemade delight!