Preheat the oven to 225°F / 105°C (90° fan-assisted) and line a large cookie sheet with parchment paper.
Add the egg whites and cream of tartar to the bowl of a stand mixer and whip on low speed for 1-2 minutes until they become foamy and opaque. Starting on low speed helps to develop smaller air bubbles for a more stable meringue.
Gradually increase the speed to medium-high, then start adding the sugar, one tablespoon at a time, letting it mix in for 15-30 seconds in between each spoonful to allow the sugar to fully dissolve.
Once all the sugar has been added, whip for another 30 seconds, then test the meringue to make sure the sugar has dissolved by rubbing some of it between your fingers - it should feel completely smooth. If it feels gritty at all, keep whipping.
Scrape down the sides of the bowl and then turn the mixer up to high speed and whip until stiff, glossy peaks form. This means that the peaks stand up straight with no drooping.
Transfer the meringue to a piping bag with a star piping tip and dot a small blob of meringue on the underside of each of the four corners of the parchment paper, to help it stick to the baking sheet.
Pipe your meringue out in any shape you like. I like to do swirls or roses. Alternatively, you can skip the piping bag and use a spoon to just scoop mounds of meringue onto your baking sheet for a more rustic look.
Bake for 1½ to 2 hours, until the meringue cookies are completely dry. When they are fully baked, they should feel dry to the touch and lift away from the parchment paper easily (check this without taking them out of the oven).
Turn the oven off and leave the meringue cookies inside with the oven door shut to allow them to cool gradually. This will dry them out slowly and avoid shocking them or cracking.
Notes
It's best to use room-temperature egg whites as they will whip up better, but separating eggs is easier when they are cold. I recommend separating first and then letting the egg whites come to room temperature. Don't throw away your egg yolks! Keep them for another recipe like custard pie or pastry cream!