In a bowl, whisk eggs until light and fluffy.In another bowl, mix the flour with a drop of milk to form a paste. This will help incorporate the flour into the milk and egg mixture and prevent lumps from forming.Add flour mixture, remaining milk, rum, vanilla extract and salt to eggs. Whisk until well blended. Dip each brioche slice into the mixture, letting it soak for about 20 seconds on each side. Do not leave for too long and only prepare 1 or 2 pieces at a time so that the brioche does not get too wet.
While the brioche is soaking, melt two tablespoons of butter in the hot non-stick skillet. Depending on the size of your pan and your brioche slices, place one or two pieces of brioche in the hot pan, leaving space between each piece. Cook each side for about 2 minutes or until golden brown. Repeat with the rest of the brioche.
Serve with whipped cream, maple syrup, fruit, etc.
