Heat the butter in a pot over medium heat. As soon as it melts (do not let it burn), add the flour. Beat with a hand whisk and sauté the flour for a few minutes.
Add the milk, in small batches, whisking continuously so that no lumps form in the mixture.
The mixture will be quite thick at first. You need to be persistent and patient. As you add more and more of the milk, you will see how the béchamel sauce will start to become nice and creamy.
When the béchamel sauce starts to thicken, you have to be very careful not to let it burn on the bottom. Whisk continuously.
When ready, remove from heat and add salt and pepper. Whisk in order to reduce temperature.
At the end add the 3 egg yolks and whisk to incorporate.