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1. Make filling: Bring 1/4 cup sugar and 1/4 cup water to a simmer in a small saucepan over medium heat, stirring until sugar has dissolved; set aside.
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2.Put egg yolks in the heatproof bowl of an electric mixer set over a pan of simmering water. Whisk by hand until yolks are warm to the touch, about 2 minutes. Fit mixer with the whisk attachment. Mix yolks on medium speed; pour in sugar syrup. Raise speed to high; mix until mixture is cool and fluffy, about 3 minutes. Reduce speed to medium; drizzle in melted chocolate. Refrigerate until firm, about 45 minutes.
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3.Make cookies: Put pistachio paste and rest of sugar in the clean bowl of an electric mixer fitted with the paddle attachment. Mix on medium speed until smooth. Mix in the egg white, vanilla, and salt.
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4. Preheat oven to 350 degrees. Transfer dough to a pastry bag fitted with a small (about 1/4 inch) plain round tip (such as Ateco #10). Pipe 1-inch rounds onto a baking sheet lined with parchment paper, spacing about 1 inch apart.
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5. Bake cookies until firm and golden around edges, about 10 minutes. Transfer on parchment to a wire rack, and let cool completely.
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6. Place cookies, bottom sides up, on a wire rack set over a baking sheet. Spread 1 to 1 1/2 teaspoons chocolate filling on bottom of half of the cookies. Sandwich with remaining cookies. Freeze 1 hour.
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7. Make glaze: Melt chocolate in a heatproof bowl set over a pan of simmering water, stirring occasionally. Remove from heat. Add shortening; stir until combined. Let cool, stirring, until mixture is lukewarm.
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8. Using a chocolate fork or regular fork, hold 1 cookie at a time above bowl of chocolate glaze. Spoon glaze over cookie, letting excess drip back into bowl. Transfer to a wire rack set over a baking sheet. Garnish with slivered pistachios. Refrigerate until set, 5 minutes. Cookies can be refrigerated in single layers in airtight containers up to 3 days.
