Preheat oven to 400 degrees F. Grease 18 muffin cups or line with paper muffin liners.
Combine flour, sugar, 1 cup Andes mints, cocoa powder, baking soda and salt in a large bowl. Whisk egg, sour cream, milk, vanilla, and vegetable oil in another bowl until smooth; pour into chocolate mixture and stir until batter is just blended. Fill prepared muffin cups about 2/3 full and sprinkle with remaining 1/4 cup chopped Andes mints.
Bake in preheated oven until a toothpick inserted into the center comes out clean, about 15 - 18 minutes. Cool in the pans for 10 minutes before removing to cool completely on a wire rack.