Make ganache first: Heat the cream in a small saucepan over medium until steaming with small bubbles at the edges (do not boil). Pour over the dark chocolate chips (and corn syrup, if using) in a heatproof bowl. Let sit 2 minutes, then whisk until smooth and glossy. Cover and refrigerate 25–30 minutes to thicken to a pipeable consistency.
Preheat the oven to 350°F. Lightly grease a 24-cup mini muffin pan or line with mini paper liners for easy release.
Brown the butter: In a light-colored saucepan, melt the butter over medium heat, stirring, until the milk solids turn golden and it smells nutty, 5–7 minutes. Immediately scrape into a large mixing bowl and cool 10 minutes.
Whisk the sugars into the cooled brown butter until it looks like wet sand. Add eggs one at a time, whisking well after each. Stir in vanilla and peppermint extracts.
In a separate bowl, whisk together flour, pudding mix, cocoa powder, baking soda, salt, and espresso powder until evenly combined.
Switch to a spatula. Add the dry mixture to the wet in two additions, stirring just until the flour streaks disappear. Stir in the cooled brewed coffee to bring the dough together. Fold in the chopped Andes mints. Chill the dough 15 minutes to firm slightly.
Scoop about 1½ tablespoons of dough into each mini muffin cup. With a floured thumb or the back of a 1-teaspoon measure, make a shallow indent in the center (do not press through).
Bake for 12–13 minutes, rotating the pan halfway, until the edges are set and the tops lose their shine.
Immediately reinforce the wells: Press the center again with a clean, lightly oiled teaspoon or a pastry tamper. Cool in the pan 8 minutes, then carefully twist each cup to release and transfer to a rack to cool completely.
Fill and finish: Spoon or pipe the cooled ganache into each cookie cup. Sprinkle with chopped Andes mints. Let stand 45–60 minutes to set before serving.