First make a roux: leave out the butter in a saucepan. Sprinkle in the flour and stir well. Pour in the milk immediately. Stir-stir-stir until you get a nice sauce. If necessary, add a little more milk if the sauce is too thick. Stir in the mustard.
Depending on how hot it should be, you choose the mustard, from medium hot to hot or Dijon mustard (which is particularly spicy). If you like it particularly mild (or for children), you can use the medium-spicy with Danish. Mix the hot dog mustard (it has a mild, sweet taste). Season to taste with salt and pepper.
In the meantime, boil the eggs (I prefer the egg yolks still soft on the inside, about 5.5 minutes).
Leave the bacon in a pan and pour over the finished sauce, put the eggs in it.
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