Preheat the oven to 325°F (163°C). Line two baking sheets with parchment paper.
In a medium bowl, whisk together the all-purpose flour, cornstarch, and salt. Set aside.
In a large bowl, beat the softened unsalted butter and powdered sugar together with an electric mixer on medium speed until light and fluffy, about 3 minutes.
Add the vanilla extract and mix until combined.
Gradually add the flour mixture to the butter mixture, mixing on low speed until just combined and a soft dough forms.
Divide the dough into three equal portions.
For the classic shortbread cookies: Roll one portion into a log about 2 inches in diameter. Wrap in plastic wrap and refrigerate for 15 minutes.
For the lemon zest shortbread cookies: Add the lemon zest to the second portion of dough and gently knead until evenly distributed. Roll into a 2-inch diameter log, wrap in plastic wrap, and refrigerate for 15 minutes.
For the chocolate-dipped shortbread cookies: Roll the third portion into a 2-inch diameter log, wrap in plastic wrap, and refrigerate for 15 minutes.
Remove the dough logs from the refrigerator. Using a sharp knife, slice each log into 1/4-inch thick rounds and place them 1 inch apart on the prepared baking sheets.
Bake the cookies in the preheated oven for 15 to 18 minutes, or until the edges are lightly golden.
Remove the cookies from the oven and let them cool on the baking sheets for 5 minutes, then transfer to a wire rack to cool completely.
For the chocolate-dipped variety: Melt the chopped semi-sweet chocolate in a microwave-safe bowl in 20-second intervals, stirring between each, until smooth.
Dip half of each chocolate-dipped shortbread cookie into the melted chocolate, allowing excess to drip off. Place the dipped cookies on parchment paper and let the chocolate set at room temperature for about 30 minutes.
Store all varieties of shortbread cookies in an airtight container at room temperature for up to 1 week.