In a large pot, arrange the eggs in a single layer. Fill the pot with water until it is about ½-inch above the eggs.
Bring the water to a boil, then cover the pot and remove it from the heat.
Let the eggs rest in the covered pot for 14 minutes. While the eggs are cooking, prepare a large bowl of ice water.
After 14 minutes, use a slotted spoon to transfer the eggs to the ice water. This will stop them from cooking further. Allow them to rest in the ice water for 4-5 minutes.
Peel the eggs under water and pat dry with a paper towel. Using a sharp knife, halve the eggs lengthwise. Use a small spoon to gently remove the cooked yolks. Add the yolks to a medium bowl and set the egg whites aside.
To the yolks, add mayonnaise, yellow mustard, hot sauce, apple cider vinegar, and kosher salt. Mash the yolks with a fork and then whisk to combine the ingredients until smooth.
Set the filling aside while you prepare the shrimp.