Preheat oven to 375 F.
Using flour as needed, roll out puff pastry to 1/8 inch thickness. Drape pastry dough over a 4 inch tart pan. Press dough into pan and remove excess dough. Repeat process with remaining dough and tart pans, re-rolling dough as necessary. Prick tarts with a fork, loosely cover with aluminum foil and top with pie weights. Transfer tarts to oven and bake for 15 minutes.
Remove pie weights and aluminum and bake for an additional 5 minutes, until crisp and golden. Place tarts on a cooling rack.
Once cool, fill tarts with ice cream and freeze for 1 hour.
Meanwhile, in a small pan on medium heat combine blueberries, sugar and rum. Cook for 10 minutes, until blueberries soften and let out some juice.
Remove tarts from freezer and top with rum blueberry mixture. Freeze again until ready to serve. When serving, allow pies to thaw enough for the berries to soften.