Baby Bûche de Noël Cookies

In a medium bowl, using an electric mixer at medium speed, beat the butter with the granulated sugar, brown sugar, salt, and vanilla until smooth. Add the flour and 1/3 cup cocoa and beat at low speed just until incorporated, then beat in 2 tablespoons of cream until smooth. Turn the dough out onto a work surface and roll it into two 16- inch-long ropes. Wrap the dough in plastic and refrigerate until firm, about 20 minutes. Preheat the oven to 400 degrees F and line a baking sheet with parchment paper. Cut the cookie dough into 2-inch logs and arrange them on the baking sheet. Bake for 20 minutes, until the tops crack lightly and the cookies are firm to the touch. Transfer the baking sheet to a rack and let the cookies cool completely. In a medium bowl, using an electric mixer at medium speed, beat the softened cream cheese with the confectioners' sugar, 1/2 teaspoon cocoa, and 1 1/2 teaspoons heavy cream until smooth. Beat in the melted chocolate. Spread the frosting on each of the cookies in lengthwise strips to resemble the bark of a tree. Return the cookies to the baking sheet and let stand at room temperature until the frosting firms up, about 2 hours.