Combine butter, sugar, and water in a medium saucepan and cook over low heat, stirring occasionally, until butter is melted, about 3 minutes. Add chocolate chips and stir constantly until chips are melted. Transfer mixture to bowl of stand mixer fitted with paddle and let cool for 10 minutes.

Whisk flour, baking soda, and salt together in second bowl. Add eggs to bowl with chocolate mixture and beat on medium-high speed until smooth, about 2 minutes. Reduce speed to low, add flour mixture in 3 additions, and mix until just combined. Cover bowl with plastic wrap and refrigerate dough for 1 hour.

Adjust oven racks to upper-middle and lower-middle positions and heat oven to 350 degrees. Line 2 baking seets with parchment paper. Working with 1 heaping tablespoon dough at a time, roll into balls; space balls 2 inches apart of prepared sheets. Bake until just set, 7-9 minutes, switching and rotating sheets halfway through baking. immediately place 1 minute in center of each cookie; let stand until mint is softened, about 5 minutes, then spread melted mint over tops of cookies. Transfer cookies to wire rack. Repeat with remaining dough and mints. Let cookies cool completely.