Preheat the oven to 350°F. Lightly grease a 13-by-9-inch baking pan and set it aside.
Make the crust: In a medium bowl, combine 1 box devil's food cake mix, 1 egg, and 8 Tbsp melted butter. Beat with an electric mixer on medium until combined and evenly moistened.
Press the cake-mix mixture firmly and evenly into the bottom of the prepared 13-by-9-inch pan.
Make the filling: In a large bowl, beat 1 (8-oz) package softened cream cheese until smooth. Add 1 cup Nutella and beat until combined and smooth.
Add 3 eggs, 1 tsp vanilla, and the remaining 8 Tbsp melted butter to the cream cheese–Nutella mixture. Beat until blended.
Add the entire 1 (16-oz) box powdered sugar and mix on low to medium speed until the filling is smooth and fully incorporated, scraping down the sides of the bowl as needed.
Spread the Nutella–cream cheese filling evenly over the pressed cake-mix crust.
Bake in the preheated 350°F oven for 40 to 50 minutes, until the edges are set and the center remains slightly gooey. Do not overbake.
Remove the pan from the oven and let the cake cool on a wire rack. Allow it to cool until it is set enough to slice (cooling completely will make slicing cleaner). Slice and serve.