Preheat oven to 175 degrees. Bring sugar and 2/3 cup water to a boil in a small saucepan. Swirl pan to dissolve sugar. Wash down sides with a wet pastry brush to prevent crystals from forming. Boil until syrup registers 238 degrees on a candy thermometer.
Put egg whites into the bowl of an electric mixer fitted with the whisk attachment; beat on low until foamy. Add cream of tartar. Raise speed to medium-high. Beat until stiff (but not dry) peaks form.
With mixer running, slowly pour in hot sugar syrup. Raise speed to high. Beat until cooled, about 10 minutes. Place one-third of meringue in each of 2 bowls. Fold vanilla into 1 bowl. Fold 3 tablespoons coffee extract into other bowl. Sift cocoa over the mixer bowl; add chocolate extract and remaining 2 1/4 teaspoons coffee extract, and beat on high speed until combined.
Transfer each batch to a separate pastry bag fitted with a 1/2-inch plain round tip (such as Ateco #806). Pipe 1-inch drops onto baking sheets lined with parchment, spacing about 3/4 inch apart. Bake until hard to the touch but not browned, 5 hours or overnight. Transfer meringues to wire racks; let cool completely. Meringues can be stored in airtight containers up to 1 week.
Melt chocolate in a heatproof bowl set over a pan of simmering water, stirring occasionally. Remove from heat. Stir in oil. Dip bottoms of meringues in melted chocolate; transfer to baking sheets lined with parchment paper to set, about 1 hour.
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