In a large bowl, whisk together the gluten-free flour blend, granulated sugar, and salt.
Add the softened unsalted butter and vanilla extract to the dry ingredients. Mix with a hand mixer or stand mixer on low speed until well combined and a dough forms. It will be crumbly at first, then come together.
Turn the dough out onto a lightly floured surface and gently bring it together into a ball. Flatten the ball into a disc, wrap in plastic wrap, and refrigerate for at least 30 minutes.
Preheat your oven to 325°F (160°C). Line baking sheets with parchment paper.
On a lightly floured surface, roll out the chilled dough to about 0.5 inch thickness. Cut out desired shapes using cookie cutters. Alternatively, you can form the dough into a log, chill again, and then slice into rounds.
Place the cookies on the prepared baking sheets, spacing them about 1 inch apart.
Bake for 18-20 minutes, or until the edges are lightly golden brown. The centers should still be pale.
Let the cookies cool on the baking sheets for a few minutes before transferring them to a wire rack to cool completely.
Notes
Store completely cooled cookies in an airtight container at room temperature for up to 3 days. You can also freeze the dough or baked cookies for longer storage.