Soften butter to room temperature but not melted. Sift together the flour, cocoa, cornstarch (if using), and salt into a bowl. Refrigerate the dry mix while you cream the butter.

In a large bowl, beat the softened butter, confectioners’ sugar, and granulated sugar until light and fluffy — about 3 minutes with a hand mixer or 2 minutes with a stand mixer on medium. Scrape the bowl. Add the vanilla and beat until combined.

Gradually add the dry mixture to the butter mixture on low speed or fold in with a spatula until a soft dough forms. The dough should be cohesive and slightly crumbly but hold together when pressed. If it’s too dry, add 1 teaspoon milk; if too sticky, chill 10 minutes.

For sliced shortbread logs: shape dough into a 2-inch diameter log, wrap in plastic, and chill 20–30 minutes until firm. Slice into 1/4-inch rounds before baking.

For rolled shortbread: chill dough 20–30 minutes, roll on a lightly floured surface to 1/4-inch thickness, and cut with a cookie cutter.

Preheat oven to 325°F (163°C). Arrange cookies on a parchment-lined baking sheet with 1 inch between them. Bake 18–22 minutes for 1/4-inch cookies, until edges are set and centers no longer look raw — they will still be slightly soft but firm as they cool. Rotate the pans halfway through baking for even color.

Let cookies cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely. For the drizzle, melt dark chocolate with a teaspoon of oil or butter in 30-second intervals, stirring until smooth. Drizzle or dip cooled cookies and let set at room temperature.