In a medium bowl, whisk together the flour, baking powder, and salt. Set aside.
In a large mixing bowl, cream together the vegan butter and granulated sugar until light and fluffy.
Add the unsweetened applesauce and vanilla extract to the butter-sugar mixture. Mix until well combined.
Gradually add the dry flour mixture to the wet mixture, mixing until a dough forms. The dough will be crumbly, but it should hold together when you knead it gently with your hand.
Divide the dough in half, form each half into a disk, wrap in plastic wrap, and refrigerate for at least 1 hour or until firm.
Preheat the oven to 350°F (175°C) and line baking sheets with parchment paper.
On a lightly floured surface, roll out one disk of dough to about ¼ inch thickness. Use cookie cutters to cut out desired shapes.
Place the cut-out cookies on the prepared baking sheets, leaving a bit of space between them.
Bake for 10-12 minutes, or until the edges are lightly golden. Let the cookies cool on the baking sheets for a few minutes before transferring them to a wire rack to cool completely.