First, we'll make the ice cream. You'll need to let this cake batter ice cream base chill for 4 hours in the fridge before putting into your ice cream maker, so make sure you give yourself enough time!
Grab a large bowl and fill it with 2 cups of ice and 1 cup of water. Nest another bowl into that bowl so that it's sitting on the ice water. Add cream and vanilla to the inner bowl. Place a strainer on top of the inner bowl. Set this setup aside.
In a medium saucepan, add milk, sugar, and salt. Whisk together to combine and then place over medium heat until it starts to simmer. Remove from heat. In a medium bowl, whisk egg yolks and then very slowly whisk in the hot milk. If you add too fast, the eggs will scramble. Return yolk mixture to sauce pan.
Place saucepan over medium-low heat. Stir continuously as it begin to heat up. After about 5-10 minutes, the mixture should start to thicken very slightly. Once the mixture coats the back of a spoon (it should still be a thinner mixture - not like pudding!) , remove from heat and strain into the heavy cream and vanilla mixture. This will remove any scrambled egg pieces and milk film.
Stir egg mixture into heavy cream until mixture has cooled down. Add cake mix and whisk together until lumps are gone. Place plastic wrap on top of the ice cream base so that it's touching the liquid. Refrigerate until chilled through - about 3-4 hours.
Freeze ice cream according to your ice cream maker's instructions (mine took about 17 minutes to freeze). When the ice cream is about 3/4 of the way done, add sprinkles as it's churning.
When the ice cream is finished, spread half of it into rectangular glass container with a lid. Crumble half of the brownies over the ice cream. Then finish with a drizzle of about 1/2 of the hot fudge and a few more sprinkles. Repeat layer one more time and then spread top of the ice cream into one flat layer. Place plastic wrap directly on top of the ice cream and then cover with the lid. Freeze until firm, about 4 hours.