2 cups all purpose flour
5 tablespoons unsweetened cocoa powder
1 1/8 teaspoons baking powder
1 teaspoon baking soda
1/2 cup nonhydrogenated solid vegetable shortening or unsalted butter (1 stick), room temperature
1 cup sugar
2 large egg yolks
1 tablespoon vanilla extract
1 cup whole milk
1/2 cup plus 2 tablespoons nonhydrogenated solid vegetable shortening or unsalted butter, room temperature
1/2 teaspoon peppermint extract
1/4 teaspoon vanilla extract
2 cups plus 3 tablespoons powdered sugar
1/4 cup egg whites (from about 2 large eggs)
3 to 4 drops green food coloring
3 tablespoons heavy whipping cream
1/2 cup bittersweet or semisweet chocolate chips
Pastry bag or resealable plastic bag
Medium-size star pastry tip
Position rack in center of oven and preheat to 375°F. Line 2 large rimmed baking sheets with parchment paper. Sift flour, cocoa powder, baking powder, and baking soda into medium bowl. Using electric mixer, beat shortening, sugar, egg yolks, and vanilla in large bowl until well blended, about 2 minutes. Add flour mixture in 3 additions alternately with milk in 2 additions, beating
until blended after each addition.
Drop dough by very rounded tablespoonfuls onto prepared baking sheets, spacing 3 inches apart. Bake 1 sheet at a time until cookies are slightly puffed and spread but are still soft, 8 to 10 minutes. Cool on sheets 10 minutes. Carefully transfer cookies to racks and cool completely. Repeat with remaining cookie dough. DO AHEAD Cookies can be made 1 day ahead. Store airtight in single layer at room temperature.
Using electric mixer, beat shortening, peppermint extract, and vanilla in large metal bowl until blended. Add powdered sugar, egg whites, and pinch of salt; beat until light and fluffy. Add food coloring drop by drop for desired shade of green; beat until well blended. If mixture looks curdled, place bowl over low heat several seconds and beat until smooth. Repeat as necessary
Line rimmed baking sheet with parchment paper. Bring cream to simmer in heavy small saucepan. Remove from heat. Add chocolate; whisk until smooth. Cool until lukewarm but spreadable, about 15 minutes. Spoon 1 teaspoon ganache onto bottom (flat side) of half of cookies. Place cookies, ganache side up, on prepared baking sheet.
Spoon mint filling into pastry bag fitted with medium star tip (or spoon filling into resealable plastic bag, squeezing filling into 1 corner of bag, then cut off 1/4 inch of plastic bag corner to allow for piping). Starting at outer edge of bottom (flat side) of remaining cookies and working toward center, pipe mint filling in spiral. Place 1 mint-filled cookie, mint side down, atop each ganache-topped cookie, pressing slightly to adhere. DO AHEAD Can be made 2 hours ahead. Store airtight at room temperature.