Preheat oven to 350ºF. Line an 8x8 inch baking pan with parchment paper.
In a medium mixing bowl, whisk flour, salt, baking soda and cream of tartar to combine. Set aside.
In a large mixing bowl or stand mixer, cream butter, sugar and lemon zest until light and fluffy, about 3 minutes.
Add egg, vanilla, and lemon juice. Mix until just combined.
Gradually fold dry mixture into wet mixture.
Press dough evenly into baking pan.
Bake for 16-18 minutes until the edges begin to brown.
Remove from oven and allow to cool completely.
While cookie is cooling, beat butter and powdered sugar together until smooth.
Add strawberries and lemon juice, and beat until icing turns pink and strawberries are mostly pureed.
When sugar cookie is cooled, remove from pan, and spread icing on top.
Cut into 16 bars and refrigerate until ready to serve.