Make the cake layers. Preheat the oven to 325°F (165°C). Grease three 8-inch round cake pans and line the bottoms with parchment paper.

In a medium bowl, whisk together the gluten-free flour, cornstarch, baking powder, baking soda, and salt until completely uniform.

In a large bowl, whisk together the sugar, milk, apple cider vinegar, neutral oil, yogurt, vanilla bean paste, and cake batter extract until smooth and fully emulsified.

Add the dry ingredients to the wet ingredients in two additions, whisking gently after each addition until the batter is completely smooth. Do not overmix.

Let the batter rest for 5 minutes so the gluten-free flour can hydrate. Whisk again for 10–15 seconds, just until smooth.

Fold in the sprinkles gently with a flexible spatula, stopping as soon as they are evenly distributed.

Divide the batter evenly among the prepared pans, preferably by weight. Smooth the tops and tap each pan lightly on the counter to release large air bubbles.

Bake for 24–27 minutes, or until the centers are set and the internal temperature reaches 202–205°F (94–96°C). A tester inserted in the center should come out clean or with a few moist crumbs.

Cool the cakes in the pans for 10–15 minutes. Turn them out onto wire racks, remove the parchment, and cool completely before assembling.

Make the cake batter lava. In a medium saucepan over low heat, warm the butter and heavy cream, whisking frequently, until fully melted and smooth. Remove from the heat.

Add the sifted powdered sugar and whisk until completely smooth and glossy.

Whisk in the corn syrup or glucose, vanilla bean paste, salt, and ⅛ teaspoon cake batter extract until fully emulsified. Taste once cool and add more extract only if needed.

Transfer the lava to a heatproof container. It will be quite thin when it first comes off the stovetop and will thicken as it cools. Refrigerate until completely chilled before assembly.

Make the cookie dough. Beat the butter and granulated sugar until light and fluffy. Add the vanilla bean paste, cake batter extract, and salt.

Mix in the King Arthur Gluten-Free Measure for Measure Flour until a dough forms, then fold in the rainbow sprinkles and white chocolate chips. Refrigerate for 30 minutes.

Make the cake batter buttercream. Beat the butter on medium-high speed for 3–5 minutes, scraping down the bowl as needed, until very light and fluffy.

Add the powdered sugar gradually, mixing on low after each addition. Once incorporated, mix on medium speed until smooth and airy.

Add the vanilla bean paste, salt, and ⅛ teaspoon cake batter extract. Mix until fully combined, taste, and add more extract only if needed.

With the mixer running on low, slowly stream in the room-temperature heavy cream. Mix just until the buttercream is silky and fully incorporated.

Assemble the cake. Level the completely cooled cake layers if needed. Place the first layer on a cake board or serving plate.

Spread a thin, even layer of cake batter buttercream over the cake. Pipe a tall, continuous buttercream dam around the outer edge.

Spoon the completely chilled cake batter lava inside the dam and spread it gently into an even layer. Keep the lava below halfway up the height of the dam, then scatter small chilled cookie dough pieces and a light layer of sprinkles over the filling.

Place the second cake layer on top and repeat with another thin layer of buttercream, a sturdy dam, chilled lava, cookie dough pieces, and sprinkles. Top with the final cake layer.

Apply a thin crumb coat over the top and sides. Refrigerate for 15–20 minutes, or until the buttercream feels firm.

Frost the cake with the remaining buttercream and decorate with the remaining cookie dough pieces and sprinkles as desired. Chill for another 20–30 minutes to set the structure before slicing.