Preheat the oven to 200°F/90°C with the rack in the middle of the oven.
Place the egg whites in the bowl of a stand mixer with the whisk attachment and sprinkle in the cream of tartar. Then, in a small saucepan stir together the sugar and the water until the sugar is moistened. Cook on medium heat, stirring constantly, until the sugar dissolves and the syrup is bubbling. Add a candy thermometer and cook until the syrup reaches 200°F/90°C.
Turn on the mixer to medium-low speed, while the syrup continues to cook, and whip until the whites are beginning to form loose peaks. If the egg whites are ready before the syrup is ready, turn the mixer to the lowest setting just to keep them moving.
Heat the syrup to 236°F/113°C, increase the mixer speed to medium-low, and slowly pour the syrup between the beater and side of the bowl in the stand mixer (while the beater is still mixing). Increase the speed to high and beat until the meringue holds stiff peaks. The meringue should be around room temperature. Add the vanilla and mix until incorporated.
Spoon the meringue into a piping bag fitted with a 1M piping tip (or any piping tip you would like) and pipe the meringue cookies onto a baking sheet lined with parchment paper. I love these pre-cut parchment paper sheets which lie flat rather than curl at the edges from the roll. This means your cookies will stay flat in the oven while baking.
Get creative with how you want these to look. I made peaks by holding the piping bag perpendicular to the parchment paper (i.e., straight up and down) with the tip slightly above the baking sheet and squeezing the piping bag until the cookie is the wanted size. Then, stop squeezing the bag and lift straight up to form a peak.