Prepare Ahead: Peel the large pineapple and cut in half, then into quarters down length, then cut out and discard the central core. Roughly chop then whizz in a food processor until it forms a rough purée. Tip into a bowl and stir in the lime juice, sugar and coconut cream. Put the bowl in the freezer for 2-3 hours until set about 3cm/1in in from the edges, then remove and whisk to break down the large ice crystals. Freeze for a further 1-2 hours until firm.
Meanwhile, peel the medium pineapple and thinly slice. Tip the sugar, cinnamon and 2 tablespoons water into a pan. Gently heat until the sugar has dissolved, then boil until it forms a light syrup. Add the pineapple and cook for 2-3 minutes, turning them, until they start to intensify in colour, remove from the heat and cool.
On the day: half an hour before serving, transfer the ice to the fridge. Serve with the pineapple and scatter with mint leaves and cinnamon from the poaching liquid.