Add the butter and mint to a small saucepan over medium heat. Once the butter is melted, continue to cook for 2 to 3 minutes, until the mint is very fragrant. Set aside for 30 minutes.
Use a fine mesh sieve to strain the butter into the bowl of a stand mixer. Press the mint with a spatula or the back of a spoon to release all of that butter! Add the sugars and mix on medium-low speed until creamy and fluffy, 3 to 5 minutes.
Add the egg and vanilla and mix until completely combined. In a small bowl, whisk together the flour, baking soda, salt, and ginger, if using. Add to the bowl of the stand mixer and mix on low until just combined. Stir in the chopped white chocolate, then scoop onto a sheet of plastic wrap, flatten into a disc, and chill for 1 hour.
Heat the oven to 350° F and line two sheets with parchment paper. Divide the dough into 12 equal portions, rolling each into a ball. Space 6 balls on each cookie sheet (the cookies will spread), then bake for 11 to 12 minutes, rotating the pans halfway through.
Let cool on the baking sheet for 2 minutes, and then transfer to a cooling rack to cool completely.