Line two baking sheets with parchment paper. Preheat the oven to 200 degrees Fahrenheit. Place room-temperature egg whites in the bowl of an electric mixer fitted with a whisk attachment. Beat the egg whites on medium speed.
Once they are very frothy, stop the mixer and add the cream of tartar. Start the mixer again and continue to beat the egg whites. Once they form soft peaks, increase the speed to high and gradually add the sugar, one tablespoon at a time.
Continue beating the whites until they are very shiny and hold stiff peaks, but are not dry or crumbly. Spoon the meringue into a large pastry bag fitted with a 1/2-inch round tip.
First, pipe the mushroom caps. Hold the pastry bag at a 90 degree angle, about 1/2 inch from the parchment paper. Squeeze out a round meringue disc about two inches in diameter and one inch high. Stop squeezing, then twist the bag and lift it from the meringue to get a clean break from the cap. Repeat in regular intervals on the baking sheet until you have approximately two dozen mushroom caps.
Next, pipe the mushroom stems. Again, position the bag perpendicularly, about 1/2-inch from the baking sheet. Begin squeezing the bag to form a one inch round base. Continue to squeeze as you slowly and evenly draw the bag up, then release pressure (causing a break) to form a tapering stem about two inches tall. Use the remaining meringue to pipe as many stems as possible.
Bake the meringues at 200 degrees Fahrenheit for 90 minutes, turning them halfway through the cooking time to ensure an even bake. The meringues should be hard and dry to the touch, and you should be able to easily lift one from the parchment. Turn off the oven and let them sit in the oven for several hours or overnight.