In a medium bowl, whisk together the flour, baking soda, and salt.

In a large bowl using an electric mixer on medium speed, cream the peanut butter and butter until light and fluffy, about one minute. Add the sugars and maple syrup and beat on high speed for about 2 minutes, scraping down the sides of the bowl as necessary. Reduce speed to medium and add the egg and vanilla; beat until well blended. Add the dry ingredients and stir until just combined. Using a spatula, scrape up from the bottom of the bowl to make sure all the ingredients are incorporated. Place the bowl, covered with a cloth, in the fridge for at least 30 minutes and up to 2 hours.

Using a teaspoon or a cookie spoon, scoop up generous helpings of dough, roll between your hands to form balls, and place on the baking sheets about 2 inches apart. With a fork dipped lightly in water, press the tines onto the dough to lightly flatten. Sprinkle the tops of the cookies with salt if you wish.

Place in the oven and bake for about 12-15 minutes, until cookies are lightly browned and just set. Remove from oven, let cool 2 minutes on baking sheets, then remove cookies to a rack until they cool completely.

Cookies will keep in an air-tight container for 5 days.