Preheat oven to 350 degrees, with racks in upper and lower thirds. In a medium bowl, whisk together flour, mint, ginger, lemon zest, and baking soda. In a large bowl, using an electric mixer, beat butter, granulated and brown sugars, vanilla, and egg yolk on medium until fluffy, about 8 minutes, scraping down bowl as needed. With mixer on low, gradually add flour mixture and beat just until combined.
Place coarse sugar in a small bowl. Roll rounded teaspoons of dough into balls and gently roll in sugar to coat. Place, 2 inches apart, on two parchment-lined rimmed baking sheets. Bake until edges are slightly brown, 14 minutes, rotating sheets halfway through. Transfer cookies to wire racks to cool.