6 pints (packed) mint chocolate chip ice cream, softened in refrigerator for 20 minutes
2 cups coarsely crushed chocolate wafer cookies or Oreos
2 1/2 teaspoons unflavored gelatin
4 cups chilled heavy cream
1 cup powdered sugar
Green food coloring
Heat oven to 350°. Coat a big rimmed baking sheet (17x13x1-inch) with nonstick spray. Line bottom with a big sheet of parchment paper.
Mix flour, baking powder, and salt in a standard bowl to blend; set aside. With an electric mixer, beat egg whites on medium-high speed in a large bowl until light and frothy, 1–2 minutes. Raise the speed to high; include cream of tartar. With mixer running, progressively add sugar. Beat until meringue is pliable and pillowy, is tripled in volume, and medium peaks form, 8–10 minutes.
With a spatula, gently fold dry ingredients into meringue, being careful not to deflate. Once flour is approximately incorporated, fold in butter, oil, and vanilla; continue folding until no dry clumps of flour remain. Pour batter into arranged sheet; smooth top.
Bake until cake is light golden brown and springs back faintly when pressed, 16–19 minutes. Transmit to a wire rack and allow cooling in pan for 15 minutes. Invert cake onto rack; peel off paper and allow cool completely.
Using a notched knife, halve cake cross-wise into 2 smaller rectangles. Wrap and freeze at least 1 hour or up to 5 days ahead
Filling and Assembly
Place 1 cake rectangle on a big platter or cake stand. Spread 1/2 cup ice cream equally over cake. Add crushed cookies evenly over. Spread remaining ice cream over, being vigilant to keep cookie layer equally intact. Top with remaining cake rectangle, pressing faintly to adhere. With an offset spatula, soft sides of cake. Freeze until ice cream is stiff, about 2 hours.
Pour 2 Tbsp. cold water into a small heatproof glass or metal bowl. Add gelatin over; let soften, about 10 minutes.
Decant water to a depth of 1/2" into a little skillet set over medium heat. Move bowl with gelatin to skillet; mix until gelatin dissolves, about 2 minutes. Eradicate bowl from skillet
Place frozen cream in a large bowl. With an electric mixer, beat cream as gradually adding melted gelatin. Beat until soft peaks form. Include powdered sugar and continue to beat until intermediate peaks form and whipped cream is pliable and pillowy (not stiff or grainy). Add green food coloring to attain a light mint-green hue, 8–10 drops; fold to blend.
With an offset spatula, add whipped cream over entire cake, making dramatic swirls on top. Spread sprinkles over. Freeze until cream is set, at least 1 hour. Cake can be prepared 2 days ahead. Keep frozen.