Start your caramel frosting first so that the caramel has time to cool completely.
Combine brown sugar, butter, and heavy cream in a small saucepan over medium heat.
Stir until butter is melted and all ingredients are combined.
With heat on medium-high bring liquid to a boil, stirring constantly.
Continue to boil (while still stirring continuously) for two minutes, remove from heat, stirring occasionally so that the mixture does not burn.
Allow caramel to cool for 5 minutes and then stir in vanilla and flaky sea salt.
Allow to cool completely before proceeding -- I recommend starting your cookies and then coming back to the caramel to be sure that it cools entirely.
Once mixture is cooled, gradually stir in powdered sugar until combined. If mixture is still too runny, you may add more powdered sugar (though it will stiffen as it sets). Set aside until cookies are ready to be filled.
Preheat oven to 350F (177C) and prepare cookie sheets by lining with parchment paper. Set aside.
Combine dark chocolate chips and butter pieces in a large microwave-safe bowl.
Microwave for 30 seconds. Stir well, microwave for another 15 seconds and then stir well again. Repeat at 15 second intervals until chocolate and butter are completely melted and well-combined.
Set aside and allow to cool (about 10-15 minutes). Meanwhile, whisk together flour, baking powder, and salt.
In the bowl of a stand mixer fitted with a whisk attachment, combine eggs, egg white, and sugar and stir until combined.
Gradually turn stand mixer up to high speed and whisk for three minutes (mixture will be pale yellow in color and should be slightly thick.
Add chocolate to the mixture and use a spatula to stir together until evenly combined.
Stir in flour mixture (do not overmix). Mixture will be glossy.
Cover cookie dough batter with cling wrap and allow it to sit for 20 minutes -- the batter will thicken as it sits.
Once 20 minutes has passed, drop cookie dough by 1 ½ rounded teaspoon onto prepared cookie sheet, spacing at least 2" apart.
Bake on 350F (177C) for 10-12 minutes (tops should be set).
Allow cookies to cool completely before removing from cookie sheets and filling -- Cookies are rather delicate so handle with care.
To fill, either spread or pipe (I used a Wilton 2D tip) the frosting onto the bottom of one cookie, top with the bottom of another to make a sandwich. The frosting recipe is very rich and intentionally does not make a lot, so use it sparingly in your cookies - I like to do a ring of frosting around the edge of each cookie and then just a dab in the center and that is plenty for me!
If desired, prepare dark chocolate melting wafers according to instructions on package and use a spoon to drizzle over your cookie sandwiches. Allow chocolate to harden before enjoying.