Position racks in the upper and lower thirds of the oven and preheat to 250 degrees F. Line 2 baking sheets with parchment paper. Fit a large pastry bag or large resealable plastic bag with a star tip.
Beat the egg whites and cream of tartar in a large bowl with a mixer on medium speed until foamy. Increase the speed to medium high and beat until thick and opaque, about 1 more minute. Gradually add the sugar and beat until stiff shiny peaks form. Beat in 3/4 teaspoon vanilla. Transfer half of the meringue to a separate bowl; beat the espresso powder and the remaining 1/4 teaspoon vanilla into the remaining meringue.
Holding the pastry bag in your hand, use a rubber spatula to spread the plain meringue on one side of the bag, filling the bag only half full. Turn the bag around and spread the coffee meringue on the other half of the bag. Twist the end of the bag so the 2 colors squish together. Pipe 1 1/2-inch meringues on the baking sheets, about 2 inches apart. Decorate with sprinkles.
Bake, switching the pans halfway through, until the meringues are firm, about 2 hours. Turn off the oven and leave the meringues inside until dry and crisp, about 1 more hour.