Cut carrot into small cubes. Slice ginger. Chop pineapple slices. Keeping aside a few pieces for garnish, put the rest in a blender jar.
Add carrot and ginger, add pineapple juice and blend till smooth. Add ice and honey and blend some more.
Pour into stemmed glasses. Put a thin slice of lemon on the rim of the glasses along with a pineapple piece. Place a spring of fresh mint over them and serve immediately.