Preheat your oven to 350°F (175°C). Beat the room-temperature salted butter with powdered sugar until creamy, then stir in vanilla extract. Sift together flour and a pinch of salt, then combine with the butter mixture until dough forms.

Press the dough into a tart pan evenly. Chill for about 15 minutes. Line with parchment paper and fill with weights before pre-baking for 20 minutes. Remove weights and bake uncovered for another 20-25 minutes until lightly golden. Cool completely.

Muddle fresh mint leaves with lime juice to release their flavor, then strain. Whisk together sweetened condensed milk, egg yolks, and cream cheese until smooth. Incorporate strained lime juice, lime zest, and rum.

Pour the filling into the cooled crust, spreading evenly. Bake for about 15 minutes until set but slightly jiggles in the center. Allow cooling at room temperature before refrigerating.

Refrigerate for at least 2 hours or until fully chilled. Serve slices garnished with whipped cream, fresh mint, and lemon slices.