Add the butter and chocolate chips to a small sauce pan, and heat over medium low heat. Stir often until both are completely melted and you can stir them smooth. Set aside to cool for about 10-15 minutes.
In a large sized bowl, combine the eggs, vanilla extract, and brown sugar until smooth. Set aside.
Add the slightly cooled, melted chocolate to the egg mixture, and beat together with a stand mixer or hand mixer.
Slowly pour the dry ingredients into the wet mixture, while stirring, until combined.
Cover the dough with plastic wrap and refrigerate it for at least 1 hour.
Preheat the oven to 350 degrees Fahrenheit. Line a cookie sheet with parchment paper or a silicone liner.
Roll the dough into 1 1/2 inch sized cookie dough balls and place them on the prepared baking sheet, about 2 inches apart. (I like to do rows of 3 across, 4 down.)
Bake the cookies for 8 to 10 minutes until the edges are set and the tops just start to crack.
Remove the cookies from the oven and let them cool for 2 minutes.
Top each cookie with an unwrapped Andes mint. Let the mint set for about 1 to 2 minutes.
As the mint melts, use a butter knife to spread the chocolate over the top of each cookie, not quite to the edges.
Let the cookies cool until the chocolate on top of them is set.