Heat oven to 350°F (175°C). Line the bottoms of two 8-inch round pans with parchment, then lightly butter and dust the pans with unsweetened cocoa powder.
Place the 1/4 cup chocolate chips and the 1/3 cup plus 2 tablespoons unsweetened cocoa powder in a heat-proof bowl. Pour 1 1/4 cups boiling water over the chocolate and cocoa, then stir until smooth; set aside to cool slightly.
Whisk together 2 1/3 cups all-purpose flour, 1 teaspoon baking soda, 1/4 teaspoon baking powder, and 3/4 teaspoon salt in a large bowl.
In a separate bowl, beat 3/4 cup sugar, 3/4 cup light brown sugar, and 1 1/4 cups mayonnaise until combined, about 1–2 minutes. Add 2 large eggs and 1 egg yolk one at a time, mixing until just incorporated, then beat in 2 teaspoons vanilla extract.
Add the flour mixture to the mayonnaise mixture in three additions, alternating with the chocolate mixture, beginning and ending with the flour. Beat just until blended after each addition; do not overmix.
Divide the batter evenly between the prepared pans and smooth the tops. Bake 20–25 minutes, or until a toothpick inserted into the center comes out clean and the cake springs back when gently pressed.
Cool the cakes in the pans for 10 minutes, then transfer to a wire rack to cool completely.
To make the frosting, beat 1 cup softened unsalted butter until creamy. Add 1/4 teaspoon fine sea salt. Add the confectioners' sugar and milk in four additions (about 1 cup sugar and 1 tablespoon milk at a time), beating on low to combine then on high until smooth and creamy. Stir in 2 teaspoons vanilla extract.
When cakes are completely cool, slice each cake horizontally to make four layers total. Place one layer on a serving plate and spread a 1/4-inch layer of frosting. Repeat with remaining layers and frosting.
Drizzle 1/2 cup HERSHEY'S Caramel Topping over the assembled cake to finish.