Place a sauce pan over medium-low heat to warm.
Add the butter and allow it to foam.
Wait for the butter to stop foaming.
Once the butter stops foaming, add the flour.
Whisk butter and flour mixture for two to three minutes to a roux.
Ladle one cup of warm milk into the roux and whisk vigorously.
Add the next ladle of warm milk and whisk vigorously.
Continue adding milk one ladle at a time, whisking vigorously until all the milk is incorporated.
Continue to cook the white sauce for ten more minutes over low heat to ensure the flour is cooked, whisking constantly to ensure the sauce does not stick to the bottom or sides of the pot and brown or burn.
Remove from heat.
Add a pinch of nutmeg.
Add salt and pepper to taste.
Work the sauce through a fine mess strainer.
Add the eggs to make the sauce rich and luxurious.
Temper the first egg with a tablespoon of the hot sauce.
Add the tempered mixture into the sauce and whisk to incorporate.
Temper the second egg add to the sauce and whisk to incorporate.