Preheat oven to 325 degrees. Make the cookies: Whisk together flour, cocoa, and salt in a bowl. In another bowl, cream butter and sugar until pale and fluffy. Beat in egg and vanilla. Gradually add flour mixture, beating to combine. Shape dough into 2 disks, wrap each in plastic, and freeze until firm, about 15 minutes.
Working with 1 disk at a time, roll out dough to 1/8-inch thickness on a lightly floured surface. Using a 1 1/2-inch biscuit cutter, cut out rounds. (Return dough to freezer if it softens.) Space rounds 1 inch apart on parchment-lined baking sheets. Freeze until firm, about 15 minutes. Brush off excess flour.
Bake cookies, rotating sheets once, until edges are firm, 12 to 14 minutes. Transfer cookies, on parchment, to a wire rack, and let cool completely.
Meanwhile, make the filling: Place chocolate in a medium bowl. Heat cream, sugar, and salt in a small saucepan over medium heat, stirring until sugar and salt dissolve. Pour over chocolate. Let stand for 10 minutes. Gently stir until smooth. Let stand until firm enough to spread, about 15 minutes.
Spread 1 1/2 teaspoons filling on half the cookies. Top with remaining cookies. (Assembled cookies can be refrigerated for up to 2 days.)