Preheat the oven to 350 degrees F. Line an 8-inch square baking dish with foil, overhanging the edges by an inch.
For the crust: Combine the butter, sugar and lime zest in the bowl of a mixer fitted with the paddle attachment. Mix on high until light and fluffy, about 2 minutes. Scrape down the sides and add the flour and salt. Mix on low speed until the dough comes together (it should be in big, loose crumbles, not dusty). Press the dough evenly into the bottom of the foil-lined pan. Bake until light golden and set, 15 to 18 minutes. Cool completely on a rack.
For the filling: Combine the whole egg and yolks, lime zest and juice and sweetened condensed milk in a medium bowl and whisk until very smooth. Pour into the crust and bake until completely set, 18 to 20 minutes. Cool completely on a rack, then refrigerate until chilled and very set, at least 3 hours.
For the topping: Preheat the broiler.
Remove the bar in one piece using the foil as handles (loosen the corners with a paring knife, if needed). Cut into 16 squares and place on a foil-lined baking sheet, spaced about an inch apart. Sprinkle the tops evenly with the sugar. Broil until the sugar is caramelized and crisp, 1 to 2 minutes, depending on your broiler. Let cool a few minutes before transferring to a serving tray. Garnish with lime quarters and serve.