Preheat the oven to 325°F (163°C). Line a baking sheet with parchment paper.
In a large bowl, cream the softened butter and granulated sugar together until light and fluffy, about 3-4 minutes using an electric mixer.
Gradually add the all-purpose flour and salt to the butter mixture, mixing on low speed until just combined and a dough forms.
Turn the dough out onto a lightly floured surface and gently knead it to bring it together. Roll the dough out to about 1/4 inch thickness.
Use a cookie cutter or knife to cut the dough into desired shapes (about 24 cookies). Place the cookies on the prepared baking sheet, spacing them about 1 inch apart.
Bake the cookies in the preheated oven for 20-25 minutes, or until the edges are lightly golden. Remove from the oven and allow the cookies to cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.
While the cookies cool, melt the chopped dark chocolate in a heatproof bowl set over a pot of simmering water, stirring until smooth. Alternatively, melt the chocolate in the microwave in 20-second intervals, stirring between each until fully melted.
Dip half of each cooled shortbread cookie into the melted chocolate, allowing excess chocolate to drip off. Place the dipped cookies on parchment paper and let the chocolate set at room temperature until firm, about 30 minutes.
Once the chocolate is set, store the cookies in an airtight container.