You can make the batter and cream filling the day of, but I HIGHLY recommend making it the day before. It will spread out the work you need to do, and the flavors will turn out better.
To make the batter, brown the butter in a small saucepan over medium heat, making sure not to burn it. Once browned, set it aside. In a second pan, heat the milk over medium heat until steaming and then allow it to cool for 10 minutes.
In the bowl of a stand mixer, beat together the eggs, flour, sugar, and salt on medium-low speed. Once fully mixed, slowly stream in both the milk and the browned butter. Mix until fully incorporated, and make sure to scrape down the sides with a rubber spatula if necessary. Once mixed, pour the batter into a container with a spout (I used a juice pitcher) and refrigerate overnight.
To make the cream filling, put the milk and vanilla bean with scrapings in a medium saucepan and boil, then remove from heat and set aside for 10 minutes, and remove the bean. For the next step, you will need one large bowl full of ice, and a smaller bowl (that can still hold all the cream filling) that can rest inside the larger bowl.
In another medium saucepan, whisk together the egg yolks, sugar, and cornstarch. Slowly whisk in the hot milk, and then place the pan over high heat until boiling, making sure to whisk the entire time. As soon as it's boiling, remove from the heat and press the cream through a fine-meshed sieve into the smaller bowl you set aside earlier. Place the bowl into the ice bath and whisk constantly until the temperature of the cream reaches 140°F. Stir in the butter until fully melted and everything is mixed together well. Once the cream filling is cool, cover it with plastic wrap and refrigerate overnight.