In the bowl of your stand mixer, beat the eggs with the sugar at the highest speed for 10 minutes, until light and fluffy.
Meanwhile, in a medium sauce pan, heat the milk and butter together, by stirring occasionally. Your preparation is ready once the butter has melted and the milk is hot (not boiling). Set aside.
In a separate mixing bowl, whisk the flour, baking powder and sea salt for about 30 seconds. Gently fold your dry ingredients to your egg/sugar preparation. Then, combine the Hot milk/butter preparation to the rest of your batter. Add the vanilla bean paste.
Bake for 30 minutes at 320F/160C. (Every oven heats differently, monitor your cake from 25min). It is ready when the toothpick inserted comes out clean.
Once your Old-Fashioned Hot Milk Cake is out of the oven, let it cool down for about 10min. Then, unmold, serve and enjoy! Bon appétit!