Fill a 3 to 5 quart pot halfway with water and bring to a boil. One at a time, use a spoon to gently lower the eggs into the boiling water so they don’t crack. Cover the pot, turn off the heat, and set the timer for 12-15 minutes for yolks that are set but not chalky; adjust time for desired doneness.
When the eggs are done, transfer them to an ice bath for at least 1 minute.
Crack and peel the eggs. Transfer them to a bowl and mash with a fork or use an egg slicer. Mashing will get you a smoother consistency, and an egg slicer is faster.
Combine milk, mayonnaise, salt, pepper, sugar and nutmeg with the eggs and mix gently. Taste, and adjust accordingly.
To assemble your sandwiches, butter one side of one slice of bread and spread the egg salad mixture on the other slice.
Place buttered bread on top, cut the crusts off, and halve the sandwiches.
Serve immediately or keep them in an airtight container in the refrigerator for 2-3 days.