Fill a large pot halfway with water and bring to a boil. See note 1 below.
Arrange eggs in steamer basket, if using. Lower basket into water. Or use a spider strainer or large spoon to gently submerge eggs a few at a time until you've added them all.
Set a timer for 12 minutes.
When the water begins to bubble vigorously again, reduce heat to maintain a gentle simmer so the eggs don't jostle around too much.
While the eggs cook, fill a large bowl halfway with ice water.
When timer rings, pull eggs out of pot and plunge into ice water.
Cool for 15 minutes.
Peel carefully and give a quick rinse under running water to remove any remaining bits of shell.
Slice each egg in half lengthwise.
Carefully remove yolks and place into a large mixing bowl. Mash well with a fork.
Add mayonnaise, mustard, salt, and pepper and continue mashing and blending until yolk mixture is creamy.
Stir in shallot.
Using a piping bag fitted with a plain tip, a resealable plastic bag with one of the bottom corners snipped off, or a spoon, pipe or spoon the yolk mixture back into the egg halves.
Sprinkle with paprika and serve.
Video
Notes
If you have a collapsible steamer basket and would like to use it to lower the eggs into the pot, makes sure it fits snugly. A 7 ½ quart Dutch oven works well.
Make-ahead options: You can boil, cool, and peel the eggs up to three days in advance. After that, it’s up to you how to proceed. You could make the filling and keep it separate, with both filling and whites tightly covered in the fridge, up to two days in advance and assemble at the last minute. Or you could even make the deviled eggs entirely up to two days in advance and just wait to sprinkle them with paprika until serving time.
Leftovers will keep well in an airtight container in the fridge for a week, as long as they haven't been left out on a buffet for a long time.