Preheat oven to 200° F. Since for a typical home oven this will require two half-sheet pans, position two racks a little above, and a little below center. Consider switching racks and rotating the pans during the drying period if your oven conditions warrant.
Beat the egg whites in the bowl at medium speed with the cream of tartar until frothy. Increase the speed and gradually add half the sugar, then the vanilla. Add the remaining sugar over the next few minutes; the meringue will get quite stiff and shiny, and the grit of unincorporated sugar crystals should be gone.
Fit the pastry bag with the tip. Hold the bag as shown, with the thumb and finger supporting the edge of the bag. Use a rubber spatula to scoop meringue into the bag, scraping it off the spatula where your thumb supports the bag. Don't overfill the bag or it will be difficult to handle.
Unfold the end of the bag and twist it shut. Using a bit less than half the meringue, pipe around 30 mushroom stems onto a parchment lined pan: starting near the parchment, guide the pastry tip with one hand, while twisting the other end of the pastry bag to squeeze out the meringue. Vary the speed at which you lift the tip to vary the shape of the stem. Lift and twist the bag (without squeezing) to separate the stem from the tip. The pointy tips will be trimmed off after the stems are cooked.
Dust the stems with cocoa now if desired and get them into the oven. (See note below.)
Now pipe the mushroom caps. You can do this without changing the pastry tip; simply hold the tip a short distance over the parchment, and smoothly squeeze the bag; the cap will billow out nicely. Stop squeezing, lift and twist to separate the tip from the cap.
Smooth the caps as needed by gently patting them with a wet but not dripping fingertip. Dust them with cocoa as with the stems, and get them into the oven.
The mushrooms are done when they are thoroughly dry and crisp. If your oven is slightly off, this may take substantially more than two hours - check by selecting a large cap and seeing whether it is no longer gooey inside.
Melt the chocolate in a double-boiler or microwave.
Trim the mushroom stems so they have nice flat tops.
For each mushroom, take a cap and smear some melted chocolate over most of the bottom. Then attach a stem.
It may help to apply chocolate to a number of caps, then let them set so the chocolate can stiffen a bit before the stems are attached. Putting a plateful of finished mushrooms in the freezer for a couple minutes is a good way to ensure the chocolate sets completely, but beware of damaging the mushrooms with humidity or condensation.
The meringue will disintegrate if exposed to moisture. Store the mushrooms in a dry place, and don't attach to a cake too long before serving.
Applying the cocoa dusting to the mushrooms before baking them is a bit easier to do all at once because it sticks better, and is more permanent. Blowing the cocoa around is a good way to get dust on the sides, and especially near the bottoms of the stems. Beware that the cocoa may dissolve in any wet spots left from smoothing the tops of the caps, leaving a smear rather than speckling or dusting; be sure to use as little water as works when doing that. Also, you won't be able to dust under the caps, nor - should you care - the bottom surface of the stems. If you wish to have more control over the appearance, you can wait until after the mushrooms are cooked and assembled, and dust them individually.
