Divide the ice cubes between two large (16-ounce) glasses. Set aside.
Cut the mint leaves into 1/8-inch ribbons, and place in a quart-sized measuring cup, or other bowl or pitcher with pouring spout. Use a cocktail muddler to smash the mint leaves liberally, until uniformly bruised.
In whatever vessel you used to brew your coffee (I use a Melitta filter over a large mug) stir in the honey until dissolved.
Pour the sweetened coffee and milk into the measuring cup with the muddled mint. Stir for about 10 seconds to infuse the mint into the liquid.
Hold a small strainer over one of your serving glasses, and pour the coffee mixture through the strainer and into the glass. Repeat with second glass. Garnish with mint and serve.