Preheat the oven to 350 degrees. Spray the bottom and sides of a 9 x13-inch cake pan with nonstick spray.
COOKIE CRUST: Combine the butter and confectioners' sugar in the bowl of an electric mixer fitted with the paddle attachment. Beat on medium-high speed for 2 to 3 minutes, until light and fluffy. Add the 2 cups flour and salt, and mix another minute, until well combined. Transfer the dough to the prepared pan and pat out to an even thickness over the bottom of the pan. Bake the crust for 20 to 25 minutes, until lightly golden. Remove from the oven, and cool on a baking rack.
LIME TOPPING: Whisk together the eggs and sugar in a large mixing bowl. Add the 6 tablespoons flour, and whisk until just combined, then whisk in the lime juice and zest. Pour the lime topping over the cooled crust, then bake for an additional 25 minutes, or until the filling is set.
Allow the bars to cool completely. Generously dust with confectioners' sugar and cut into squares. For the neatest presentation, use a metal spatula to remove the bars from the pan. Leftovers will keep for 2 or 3 days at room temperature, or for up to a week in the refrigerator.
Cook's Note: If you cannot find fresh Key limes, you can use regular limes.