Step 1

Preheat oven to 350°F, with racks in upper and lower thirds. Line 2 baking sheets with parchment. In a bowl, whisk together flour, cocoa, baking powder, and kosher salt. Melt chocolate in a heatproof bowl set over (but not in) a pot of simmering water, or in the microwave, stirring every 10 seconds.

Step 2

Stir in shortening and vanilla to combine. Beat eggs with both sugars on medium-high speed until pale, fluffy, and doubled in volume, about 5 minutes. Gently fold in flour and chocolate mixtures, alternating in two additions each, just to combine (do not overmix).

Step 3

Scoop 1-ounce (2-tablespoon) balls of dough; transfer to prepared sheets, spaced 2 inches apart. Sprinkle tops with flaky salt. Bake, rotating sheets and rack positions halfway through, until cookies spread and dome, with small cracks beginning to form on tops, 8 to 10 minutes. Let cool on sheets on wire racks 5 minutes, then transfer cookies to racks and let cool completely.

Step 4

Meanwhile, temper ice cream in refrigerator until softened but not melting, about 20 minutes. Sandwich 1/4 cup ice cream between 2 cookies; repeat with remaining ice cream and cookies. Freeze until ice cream is firm, at least 1 hour, then transfer to a freezer bag to store up to 1 week.