Finely crush the shortbread cookies in a food processor, then mix with the melted butter. Press the mixture into the base of a 9-inch tart pan and refrigerate for 30 minutes.
Soak the gelatin sheets in cold water until softened.
In a saucepan, combine the heavy cream, sugar, and vanilla extract over low heat. Bring to a simmer, then stir in the ground pistachios until well combined.
Remove the cream mixture from the heat and stir in the squeezed-out gelatin sheets until completely dissolved.
Pour the pistachio panna cotta mixture over the cookie base and refrigerate for at least 4 hours, or until set.
Before serving, decorate the top of the tart with pistachio cream and fresh pistachios.
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