In a small sauce pan over medium heat, melt butter and bring to a gentle simmer. Add mint and let simmer for 3 minutes. Remove from heat, let sit for 30 minutes. Strain mint leaves from butter. Let butter cool for one hour.
Preheat oven to 350 degrees. Line cookie sheets with parchment paper or silicone baking mats.
Use electric mixer to cream mint butter, brown sugar, and sugar until light and fluffy, about 2 minutes.
Add egg, vanilla extract, and milk. Mix until combined.
In a small bowl, stir together flour, cornstarch, salt, and baking soda. Gradually beat into butter/sugar mixture, about 1/2 cup at a time. Mix until well combined.
Stir in chocolate chunks and chocolate chips.
Form dough into about 1 tbsp sized balls and space 2" apart on prepared cookie sheets. Cook for 9-11 minutes, or until tops are lightly browned.