Combine milk, butter, and vanilla bean paste in a medium pot and bring to a simmer. Remove from stove and cool to 100°F/38°C2, stirring occasionally to avoid a skin forming.
While milk mixture is cooling, prepare the dry ingredients. Whisk together the flour, sugar, and salt in a large bowl. Set aside.
When milk mixture has cooled, temper the egg yolks by pouring a small amount of the milk mixture into the eggs while whisking constantly. Pour the egg mixture into the saucepan and whisk to combine. Pour in the rum and whisk.
Add wet mixture to dry ingredients in thirds. Whisk gently (don’t create too many air bubbles in this step) to incorporate all dry ingredients. The finished batter should be the consistency of heavy cream.
Strain the batter through a sieve into an airtight container. Place a piece of plastic wrap directly onto the surface of the batter. This will help it prevent a skin from forming. Cover the container and refrigerate for 48 hours.