1. Sprinkle gelatin over 3 tablespoons water; let soften 10 minutes. Brush insides of six 5-ounce ramekins or custard cups (3 1/4 inches in diameter) with oil.

2. Bring cream and 1/2 cup plus 2 tablespoons sugar to a simmer in a medium saucepan over medium heat, stirring occasionally. Reduce heat to medium-low; add gelatin mixture. Cook, stirring, until gelatin and sugar are dissolved. Remove from heat. Whisk in creme fraiche and vanilla. Pour into a medium bowl set in an ice-water bath. Let cool completely, stirring occasionally. Divide mixture among ramekins. Refrigerate until set, about 3 hours, or up to 1 day.

3. Line a rimmed baking sheet with parchment, and place six 4-inch tart rings on top. Divide dough into 6 pieces. One at a time, roll out each piece into a 7-inch circle (1/8 inch thick) on a lightly floured surface. Press each into a tart ring; trim flush with top edge of ring. Refrigerate tart shells until cold, about 30 minutes.

4. Preheat oven to 375 degrees. Prick bottoms of tart shells all over with fork. Line each with parchment, and fill with pie weights or dried beans. Bake until edges are golden, about 18 minutes. Remove parchment and weights; continue baking until surfaces are golden, about 10 minutes. Let cool on a wire rack.

5. Heat berries, remaining 2 to 4 tablespoons sugar (depending on sweetness of berries), and vinegar in a skillet over medium-low heat, stirring, until juicy, about 5 minutes. Let cool slightly.

6. Unmold panna cottas: Dip ramekins in warm water; pat dry. Run a small knife around edge of each panna cotta; gently coax, and invert onto a tart shell. Top with berries and sauce.