Preheat oven to 325ºF and line baking sheet with parchment paper.
In a large bowl, cream olive oil and sugar together until light and fluffy (about 2 minutes) with a hand mixer or standing mixer. Add in vanilla, lemon juice, egg, lemon zest, and and whisk again until combined.
In another bowl, add the almond flour, arrowroot, baking soda and salt. Stir together.
Add the dry ingredients to the wet, and stir until combined into a sticky dough,
Using a tablespoon cookie scoop, scoop the dough, roll into a ball and place onto the baking sheet. Bake for 12-14 minutes until edges are slightly browned.
Remove from the oven and cool cookies for 5-10 minutes before transferring to a cooling rack. These cookies are very delicate so will break if you transfer them while still hot.
While cookies are cooling, make lemon glaze by adding powdered sugar and lemon juice in a medium bowl. Stir to combine. You may need to add in more lemon juice if the glaze is too thick (start with one teaspoon at a time). Glaze should be somewhat thick. Frost each cookie with the lemon glaze. Sprinkle additional lemon zest on top if you prefer.
Let cookies sit for about 5 minutes until icing has set and serve.