Heat the butter, xylitol and cardamom in a large saucepan over medium-high heat. Slice the pears into ½-inch wedges. Once the butter has melted add the pears and allow to caramelize for 4 minutes on each side. Remove from heat and arrange in a pie plate or 3-4 cup casserole dish. Reserve about 2 tbsp purée of xylitol and pour the remaining over the pears (keep remaining in the sauce pan and set aside).
In a small bowl, whisk the eggs, egg yolks, coconut cream, purée of xylitol, rum and vanilla until combined. Pour mixture over the pears and bake for 15-20 minutes until golden brown and custard has set. Remove from oven and allow to cool slightly.
Using a pastry brush, brush the top with reserved a purée of xylitol.