In a large bowl, mix the flour, powdered sugar, vanilla extract, lemon zest, and salt. Add the vegan butter and plant-based milk. Mix, using a stand mixer or hand mixer, into a smooth dough.
Form the dough into a ball, cover with plastic wrap, and refrigerate for 1 hour. → The dough can be refrigerated for up to two days. If so, let it sit for 15 minutes at room temperature before rolling it out.
Preheat the oven to 350°F (180°C).
Place the dough on a floured surface and roll it out, using a rolling pin, to about ¼-inch (½ cm) thickness.
Using a linzer cookie cutter, cut the dough into shapes. Re-roll the remaining dough and continue cutting until all dough is used. → Make sure you have an equal number of tops and bottoms.
Transfer the cookies to a parchment paper-lined baking sheet and bake in the preheated oven for 8-10 minutes.
Remove from the oven, allow cookies to cool for 5 minutes on the baking sheet, and then transfer to a cooling rack to cool completely.
Spread ½ tsp jam on the bottom pieces. Place the tops on and lightly press them together.
You can sprinkle the finished Linzer cookies with powdered sugar before serving.