Thoroughly wash and dry the persimmon and place it in the freezer overnight, either on a plate or in a container.
When ready to use, cut off the top of the persimmon and cut the fruit into cubes while it is still firm. You can peel it if you like; that's personal preference. Add the pieces to a small bowl with the lime juice, cover with a plate or dish towel, and let it thaw on the counter. This will take about 1 hour, but see the Note below for how to speed up the process.
Once the persimmon has thawed, sprinkle in the cardamom and gently mash with a fork, easing the persimmon down into big, jammy, spreadable pieces.
Add the pistachios to a small bowl, add the lime zest, tear in the mint, and toss.
Spread a thin layer of labneh on the toast. Spoon the persimmon mixture on top, drizzle with the honey, and generously garnish with the pistachio and mint.
NOTE: To speed up the thawing process, fill a bowl with boiling water, then empty the bowl and add the persimmon. Cover the bowl and wait for 10 minutes for the persimmon to thaw.