In a mixing bowl, beat together the butter and sugar until light and creamy. Add in the egg yolk and vanilla, blending until smooth.

In a separate bowl, whisk together the flour and salt. Gradually add the dry mixture to the creamed butter mixture, alternating with the milk, until a soft dough forms.

Divide the dough into two equal portions. Stir the melted chocolate into one half until fully incorporated, creating a chocolate dough. For a uniform texture, kneading lightly with your hands may help. Refrigerate briefly to firm up slightly.

Roll each dough portion on a lightly floured surface into a rectangle, approximately 10x12 inches. Trim any uneven edges to ensure the dough layers match in size.

Place the plain dough sheet on top of the chocolate one, slightly offset so that one longer edge extends about ½ inch beyond the chocolate layer.

Starting from the edge opposite the overhang, roll the dough tightly into a log, similar to a jelly roll. Wrap in wax paper and refrigerate for at least 4 hours, or overnight for best results.

Preheat the oven to 350°F. Using a sharp knife, slice the chilled dough into ¼-inch rounds, wiping the knife clean between cuts. Arrange the cookies cut-side down on a greased or parchment-lined baking sheet.

Bake for 8 to 10 minutes, or until the edges are set. Allow the cookies to cool briefly on the baking sheet before transferring them to a wire rack to cool completely.