First, let's make the shortbread. Preheat oven to 350F. Line the bottom of a 10" round pan with a piece of parchment paper and grease the sides using vegan butter or baking spray. Set aside for now.
In the bowl of a stand mixer fitted with a paddle attachment, beat your softened butter for about 30 seconds until smooth and lump-free. Add in your sugar, and cream the mixture on medium-high until it is very light in color and fluffy.
Next, add in your vanilla extract and vanilla paste, and mix to combine.
Finally, add your salt and flour, and beat mixture until a thick cookie dough forms.
Transfer your dough to your prepared baking pan, and use your hands to press it down evenly. Use a toothpick or skewer to poke holes all over the surface of the shortbread. If desired, you can also cut the cookie into wedges at this point. It will meld back together while baking, but the lines do remain and it makes it easier to cut later.
Bake your shortbread in your preheated oven for 30 minutes, until lightly golden brown and fragrant. Do note that the cookie will not brown very much as compared to traditional cookies.
Let the shortbread cool slightly, then transfer to a cooling rack or plate to finish cooling off.
To make the icing, add your milk and vanilla paste to a small bowl and add the powdered sugar, a spoonful at a time, until a thick glaze forms. Think Elmer's glue consistency. When the shortbread has cooled to room temperature, drizzle the icing on top. Cut your shortbread into wedges and enjoy!