Heat oven to 325°F and arrange rack in the lower-middle of the oven. Spray and 8 x 8-inch pan with cooking spray and line with a piece of parchment paper long enough so two sides stick up about 2-inches above the sides of the pan and act as handles. Spray parchment with cooking spray.
Combine 1 cup of the flour, 1/4 teaspoon salt, and 1/2 cup of the sugar in a medium bowl; stir in butter until moistened throughout. Press dough into pan and even out using the bottom of a measuring cup. Bake until pale golden, about 20-25 minutes.
Meanwhile, sift the remaining 3/4 cup sugar and remaining 1 1/2 tablespoons flour into a medium bowl. Whisk in the 2 eggs, 6 tablespoons lime juice and 2 teaspoons lime zest until smooth. Whisk in 1-2 drops of green food coloring for a brighter green color if you'd like.
Let par-baked crust cool for 10 minutes then pour lime filling over warm crust and return to the oven. Continue baking until just set and has just a slight jiggle in the middle, about 20-25 minutes.
Set on a wire rack and let cool in pan for one hour then transfer to the refrigerator to chill at least an hour longer.
Lift the parchment paper out of the pan using the overhang on the sides. Dust with powdered sugar and cut into squares before serving. For neat squares, wipe the knife clean between each cut.