In a medium bowl whisk together the flour, baking soda, and sea salt. Set aside.
Using a stand mixer with the paddle attachment (or a hand mixer), mix together the butter and brown sugar until very smooth. Add in the egg, maple syrup, and vanilla and mix until combined.
Slowly add the dry ingredients to the dry ingredients and mix everything until just combined. Scrape down the sides as needed.
Chill the dough for at least 2 hours in the fridge. If you're going to chill the cookie dough overnight, cover the bowl and then let it sit out for 30-60 minutes before baking.
Preheat your oven to 350 degrees F. Line a cookie sheet with parchment paper or a silicone baking mat. Set aside.
Using a medium-sized cookie scoop (about 1.5 TBSP), scoop the cookie dough into balls and place them about 1.5 inches apart on the prepared baking sheet.
Bake for 12-13 minutes until the bottom edges are turning golden brown. Remove the cookies from the oven and transfer to a wire rack to cool completely - about 30 minutes.
Icing
Add the butter and maple syrup to a mixing bowl. Mix together with a hand mixer. Add the salt and powdered sugar and mix until smooth.
Drizzle as much or as little icing over the cookies as desired. You could even dip the tops into the icing, but you may need to make a double batch of icing to have enough.