Preheat the oven to 200°F (90°C).
Line two baking sheets with parchment and set aside.
Wipe a stainless steel bowl and the whisk of an electric mixer with a paper towel dampened with a little lemon juice to eliminate any trace of grease.
Put the egg whites into the bowl and beat on low speed until foamy, about 30 seconds.
Stop the mixer and add the cream of tartar.
Increase the speed to medium and beat until soft peaks form, about 1 minute.
Increase the speed to medium-high and add the sugar one spoonful at a time.
Beat on high speed until very stiff peaks form and the meringue is glossy and thick, about 3 minutes.
Add the vanilla and beat for 1 more minute.
Fit a large pastry bag with a large round decorating tip and pipe 2-inch-diameter cookies about 1 inch apart onto the prepared baking sheets.
Bake in the upper and lower third of the oven for 1 1⁄2 hours undisturbed, and then turn off the oven.
Leave the cookies inside the oven for 1 more hour to dry out.
The cookies should be crisp, but not browned or discolored.
Transfer the baking sheets to wire racks to cool completely.
In a heatproof bowl, melt the white chocolate either in the microwave in 20-second intervals (be careful not to burn it) or over a small saucepan of simmering water.
Line two baking sheets with fresh parchment paper and fill a shallow dish with the sprinkles.
Dip the bottom of each meringue into the white chocolate, and then immediately dip them into the sprinkles.
Put them on the prepared baking sheets and let them set, about 1 hour.
The meringues will keep in an airtight container at room temperature for up to 5 days.