Preheat oven to 350 degrees. On a lightly floured work surface, roll out dough to an 1/8-inch thickness. With a 3 1/2-inch cookie cutter, cut dough into 24 rounds (reroll scraps if necessary), then press into 24 standard muffin cups (dough should come 3/4 inch up sides). Freeze until firm, 10 minutes (or up to 1 day).
Bake until shells are golden, 18 minutes. Let cool completely in pan.
In a medium heatproof bowl, combine chocolate and 1/2 teaspoon salt. In a small saucepan, bring cream to a bare simmer over medium-high. Immediately pour over chocolate, add vanilla, and let sit 1 minute. Stir gently until mixture is completely smooth. Pour chocolate into cooled shells and refrigerate until set, 20 minutes. To serve, sprinkle with coarse salt.
Wrap cooled unsalted tartlets in plastic and keep at room temperature, up to 1 day.