Preheat oven to 350 degrees with rack in middle. Set eighteen 2 3/4-inch flan rings on a baking sheet lined with parchment paper. Make parchment collars: Line each flan ring with 9-inch-long-by-2-inch-high parchment strip, overlapping slightly; set aside.
Mix cookie crumbs, butter, and 3 tablespoons sugar in a small bowl. Divide mixture among flan rings (about 2 tablespoons per ring). Press down crumb mixture to create a flat base. Bake until set, about 13 minutes. Transfer to a wire rack to cool.
Reduce oven temperature to 300 degrees. Melt chocolate in a heatproof bowl set over a pan of simmering water. Stir until smooth. Let cool slightly; set aside.
Put cream cheese in the bowl of an electric mixer fitted with the paddle attachment; beat on medium speed until lightened, about 3 minutes. Add remaining cup sugar and the salt; mix 1 minute. Add sour cream, cocoa powder, vanilla, and melted chocolate; mix until well combined. Reduce speed to low. Add eggs and egg yolks, 1 at a time, until just combined. Divide filling among flan rings. Bake until still slightly wobbly in center, 15 to 18 minutes. Transfer to a wire rack to cool. Refrigerate, uncovered, at least 6 hours or up to 24 hours.
Carefully remove rings; peel off paper. Serve garnished with chocolate curls.