Preheat oven to 325°F. You will need a 8-inch square baking pan.
Place the shortbread cookies into the bowl of a food processor fitted with a metal blade. Process until finely ground. Melt the butter. Use a brush to brush the bottom and sides of the pan with some butter and then press in a piece of parchment paper to fit the bottom and go up the sides. Brush the top of the parchment. Add the cookie crumbs to the remaining butter and mix. Dump into the prepared pan and press to cover the bottom of the pan. Set aside.
Place the cream cheese into the bowl of a stand mixer or blender and beat on medium-high speed until smooth. Scrape down the bowl. Add the eggs, one at a time, and scrape down the bowl each time to make sure all the cream cheese and eggs are being mixed together. Add the sugar, lime zest and juice, and mix on medium speed until combined. Make sure to mix the bottom of the bowl to ensure the mixture is entirely smooth and scrape off any zest that sticks to the beater.
Pour the cheesecake batter on top of the crust. Bake for 45 minutes or until set. Let cool. Place in the fridge for five hours or overnight.
To make the meringue topping: In a small heavy saucepan, bring the sugar and water to a boil, stirring to dissolve the sugar. Continue to cook the sugar until it reaches 230ºF on a candy thermometer. You can dip a pastry brush in water and wipe down any sugar crystals that appear on the sides of the saucepan. While the sugar is cooking, in a medium bowl, beat the egg whites with an electric mixer on high speed until stiff. When the sugar is ready, turn the mixer speed to low and then slowly pour the cooked sugar into the whites, down the side of the bowl, not directly onto the wire whisk. When all of the sugar has been poured in, turn the mixer up to medium-high and beat for one minute.
To serve, you can cut into squares and then decorate with the meringue, or decorate and then slice, as you wish. To make the dots as shown, place the meringue into a pastry bag fitted with a ¼-inch tip. Squeeze out lines or swirls. You may also scoop the meringue on top of the whole cheesecake and then slice to serve.