Cut the avocado in half lengthwise, remove seeds, peel and chop into small pieces. Peel, core and dice the honeydew melon. Wash mint leaves with lamb's lettuce and spin dry.
Put the prepared ingredients with pink grapefruit juice, lime juice and the linseed into a blender jug. Finely puree everything with a hand blender. Season to taste with some lime juice. Divide the smoothie between two glasses and serve.