In a bowl of a stand mixer or with a handheld mixer, whip together the powdered sugar and vegan butter.
Add the lemon juice, zest, and vanilla extract. Mix until combined.
In a separate bowl, whisk together the flour, baking powder, and salt.
Add the dry ingredients to the wet and mix until just combined. Using a spatula, fold the dough a few times to make sure it's combined. Place the dough between two pieces of parchment paper or silicon baking mats, flattening it into a round. Refrigerate for 20-30 mins.
While the dough rests, reduce the frozen raspberries. To a small saucepan, add the frozen raspberries. Over medium heat, mash the raspberries. Once boiling, allow to simmer, stirring constantly, over low heat. Once thickened, strain into a glass. This will not make that much raspberry reduction, so make sure to scrape down the opposite side of the strainer as you push the reduction back and forth in the strainer. Leave to cool.
Roll out the dough between two sheets of parchment or on silicone baking mats. Roll the dough until it is about a 1/8 inch thick. Whatever thickness and size you cut your cookies, they will remain that way through baking. The shortbread will not spread or rise.
Cut out as many cookies as you can in one roll and place the tray in the freezer for 2-3 minutes.This makes it easy to peel off the cookies you've cut and place on another mat. My cookies were about 1.24 inches in diameter, because we are sandwiching the cookies, you don't want to cut the cookies very large. The dough is very delicate, so I do not recommend trying to transfer the cut cookies if not frozen solid first.
Once chilled enough to handle, peel off the cut-out cookies and place on a lined baking tray. Repeat the rolling, freezing, and transferring process with the rest of the dough.
Preheat the oven to 340 degrees Fahrenheit. As the oven heats up, freeze all of the cut-out cookies. You can place them fairly close together on the baking sheets, as they do not spread in the oven. Bake each tray separately, straight from the freezer.
Bake for 8 minutes, rotate the pan, then bake an additional 3-5 minutes, or until the edges are lightly golden brown. The bottoms of the cookies should be very lightly golden, not too baked. Allow to cool completely in the pan.
While the cookies cool, make the filling. In the bowl of a stand mixer or a bowl with a handheld mixer, whip together the coconut oil and the vegan butter. Once combined well, add the vanilla, salt, raspberry reduction, and lemon juice. Mix again until fully combined.
Add 1 cup of powdered sugar and whip for 2 minutes at medium speed. Scrape down the sides of the bowl as needed. Add the other 1/2 cup and mix again. If the frosting isn't sweet enough for you, feel free to slowly add in more powdered sugar. Chill the frosting before using.
Once cookies are cooled, pull the frosting from the fridge and whip again so it is light and thick. The coconut oil helps this filling be smooth and light, but it can get runny with warmth quickly. So, whip the frosting as fast as possible and be ready to fill the cookies.
Add the frosting to a piping bag fitted with a star tip, or whatever piping tip you'd like. Place all the cookies upside down, add a small circle of filling to the middle of half of them, then sandwich the cookies. Place in the fridge to set up the frosting.
If you'd like to add more decoration, take the remaining frosting and add a splash of milk, a squeeze of lemon, or, if the frosting is runny enough from the warmth of your hand, drizzle the frosting over the sandwiched cookies and add sprinkles if you'd like. Refrigerate for 20 minutes to help the frosting set up. You can use a squeeze bottle, a Ziploc bag, or a small piping tip to drizzle on the detail. Place in the fridge again until the drizzle is set.