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You can also use an immersion blender in the mixing bowl. But with my method, you can do it all by hand with a simple whisk.
Mix the dry ingredients, flour, salt and sugar in a mixing bowl.
In a measuring jug, mix the eggs with the milk, water, and the seeds from the split vanilla bean.
Make a well in the middle of the flour and pour in little by little half the milk + egg while mixing with the whisk, swirling in the center. The flour on the sides will gradually fall to the center. Always mix in the middle.
When all the flour is incorporated and the batter is smooth and homogeneous, without lumps, add the remaining milk and mix.
Finally, add the warm but not hot butter and mix well. Then add the rum, Grand Marnier or orange blossom water. It's optional since the vanilla already provides enough flavor.
If you still have lumps, you can strain the batter through a fine sieve or blend it with an immersion blender.
Cover the bowl with a clean dishcloth or tea towel and leave the batter to rest for 2 hours, as recommended by the chef. I left it on for just an hour, as I usually do, and got excellent results. If you let the batter rest, you get fluffy crepes.