Combine butter, maple syrup, and vanilla in the bowl of a stand mixer fitted with the paddle attachment; beat until smooth and creamy. Add flour and salt; continue mixing until a firm dough forms. Flatten dough into a disc, wrap in plastic wrap, and chill for 30 minutes.
Preheat oven to 350 degrees F (175 degrees C). Line 2 baking sheets with parchment paper.
Roll dough between 2 pieces of parchment paper to 1/4-inch thickness. Cut out shapes with cookie cutters and place 2 inches apart onto the prepared baking sheets.
Bake in the the preheated oven until edges are golden, 12 to 14 minutes. Cool on the baking sheet for 1 minute before removing to a wire rack to cool completely.