Using 3 bowls, separate out the egg whites from the yolks. Use one bowl for the separation process, one to house the egg whites, and one to house the yolks. You want to do this so you don't get a drop of yolk in the whites. If even a little yolk is in the egg whites, it won't create stiff peaks.
Put a pot on the stove full of about an inch or two of water and bring it up to a boil. Pour the egg whites, sugar, and cream of tartar in a heat proof bowl and place the bowl on top of the pot of boiling water. Make sure the water doesn't reach the bottom of the bowl. You can also use a double boiler if you have one!
Mixing continuously, bring the egg white and sugar mixture up to 160℉. It will look foamy at this point and you shouldn't feel any sugar crystals once it's done.
Make sure your stand mixer bowl has no grease in it. I usually take some vinegar on a paper towel and just wipe it out just in case. Pour the egg mixture into the bowl of your stand mixer and add the vanilla.
Mix on high with a whisk attachment until stiff peaks form. Once stiff peaks form, remove mixture from the bowl and store in an airtight container in the fridge.