Place butter in the bowl of your stand mixer (you can use your hand mixer or a whisk too), sift in powdered sugar, beat on low heat initially, until the powdered sugar is mixed with the butter, then increase the speed to medium and beat until light and creamy.
Add vanilla extract and beat for another 30 seconds.
Now switch to the lowest speed and add the flour in parts. Gently bring the dough together, avoid overworking the dough. If you press the dough between your palms it should hold its shape. If the dough feels too dry and crumbly, add a teaspoon of milk or more as required just until the dough comes together.
Divide the dough into two parts. Place a cling wrap on the work surface, transfer one part of the dough over the plastic wrap and shape it into square or a round disc, about 1 inch thick and wrap in the cling film. Refrigerate it for 30 minutes.
Preheat the oven to 350 degrees F. Line two baking sheets with Silpat or parchment paper. Take one part of dough, roll it out between two parchment or wax papers to 1/4 inch in thickness.
For the finger-shaped cookies: Snip off the edges to make a neat rectangle or square. Score the shortbread dough to make finger shapes. Indent the cookies with a fork or skewer.
For the round or other shaped cookies: Use your favorite cookie cutter, and make cut-out after rolling out the dough to 1/4 inch thickness. Gather the scraps, form a disc, refrigerate for another 15 minutes, reroll and make more cutouts.
Lay the cookies on the sheet at least 2 inches apart (If the cookie dough seems too soft at this stage, place the tray with the cutouts in the refrigerator for another 30 minutes. This prevents the cookies from spreading too much while baking).
Bake for about 15 minutes or until the edges begin to turn golden brown. Do not let the cookies turn brown. Smaller shaped cookies will take less time to bake, so keep an eye. Remove the baking sheet from the oven. Let the cookies stand on the baking sheet for about 2 minutes, then transfer to a wire rack and let them cool completely.
For chocolate covered shortbread: Place the chocolate chips or chunks along with the butter in a microwave safe bowl. Microwave on high power in 30-second intervals, stirring with a clean, dry spoon after each interval. It took me a total of 1 & 1./2 minute to melt the chocolate chunks. Keep stirring to make it smooth and glossy.
Dip one end of a cookie in the melted chocolate, shake off excess, sprinkle pistachio pieces and place on a baking tray lined with parchment paper. Repeat the same with the rest of the cookies.
Place the tray in the refrigerator for about 15 minutes or until the chocolate hardens and firms up. Store in an airtight container for up to 2 weeks.