Melt butter with mint leaves over medium heat in a small skillet or heavy-bottomed pot.
Continue to heat the butter, stirring periodically, until it turns brown (10-15 minutes). The butter will first boil with big bubbles (it's not done then). Then, the bubbles will get smaller and the whole thing will look more brown (that's when it's done).
Remove butter from heat and strain the mint leaves out. Press the butter out of the them as best you can. Don't worry if you don't get it all - it's impossible You only need 1/2 cup of the mint-flavored butter in the cookies.
Cool the butter to room temperature - about thirty minutes.
Preheat oven to 350 F.
In a medium-sized mixing bowl, mix mint butter (there should be about 1/2 cup) and sugar until light and fluffy.
Mix in egg white and white chocolate.
In a separate bowl, whisk together flour, baking soda, baking powder, and salt.
Add flour mixture to butter mixture a little bit at a time until just combined.
Place teaspoon-sized balls of dough about 1 1/2 inches apart on a parchment- or Silpat-lined cookie sheet. The dough will be a little crumbly, so just push it together with your hands if you need to.
Bake for 7-9 minutes or until the edges begin to brown.
Transfer to a cooling rack to cool.