Cook the eggs one of two ways.
1) Cook the eggs in an Instant Pot for 5 minutes, depressurize for 5 minutes, and then rest them in ice water for 5 minutes.
2) Or place the eggs in a pot. Add enough water to cover them by about 1 1/2 inches. Bring the water to a full rolling boil. Then cover them and turn off the heat. Let them sit for 12-15 minutes to cook. Then submerge them in ice water for 10-15 minutes to cool.
Peel the eggs. (See the tips in Notes for easier peeling.) Remove the yolks from the egg whites, keeping the whites intact. Set the whites aside for later. Put the yolks in a flat-bottomed bowl for blending. Add 1/3 cup mayonnaise, salt, Dijon mustard, and vinegar.
Blend with the immersion blender until creamy and smooth. Add additional mayonnaise, up to 1/2 cup total, if the mixture is not as creamy as desired. (If you do not have an immersion blender, press the yolks through a fine, wire-mesh strainer to break them apart. Then add the other ingredients and blend them with a fork or spoon until creamy.)
Remove any loose membranes from the egg whites. Fill a large bowl with water and add a few drops of pink food coloring. Stir to mix. Make the water as dark pink as desired. Darker pink water will color the eggs more quickly than light pink water, but may oversaturate them with color if left too long.
Keep a check on the color of the egg whites. Remove them from the water to drain when they reach your preferred pink color.
Add the prepared yolk mixture to the piping bag. Pipe filling into each egg white.
Store the eggs in the refrigerator until ready to serve.
Notes
Food Colorings
Artificial Food Coloring
The first way to dye eggs is to use artificial food coloring. Add a few drops of your chosen color to a bowl filled with water--in this case,pink. The more color added to the water, the less time it takes to dye the eggs. However, if you get distracted, they can over-saturate with color. I like to add less color and let the whites sit a little longer to dye. If they aren't changing color enough after a few minutes, add more coloring to the water and stir. Using food coloring is my preferred way to dye egg whites.
Natural Food Coloring
The second way to dye them is to use natural food coloring. For pink eggs, try using beet powderdissolved in water. Many people like the results when using powder or juice from beets. However, you may not get the same results as when using artificial food coloring. Sometimes the egg whites will turn brownish-red instead of pink. Other foods produce red hues, such as cherries and strawberries, but they can impart the wrong flavor to eggs.
How to Make Hard-Boiled Eggs Easier to Peel
If you cook eggs on the stovetop instead of in an Instant Pot, here are some tips that might make them easier to peel.
Use eggs that are at least 2 weeks old. Older eggs are easier to peel because their inside air cell increases when the egg gets older. The egg shrinks as the air cell enlarges, thus causing the egg to pull away from the shell.
Put 1 teaspoon of baking soda in the water before cooking to increase the eggs' alkaline pH. Fresh egg whites stick to the outer shells because of their acidic pH. Baking soda helps reduce acidity.
Place a boiled egg into a glass with water. Place your hand or a small plate over the glass. Shake the glass hard until there are tiny cracks all over it. Then gently peel away the shell.
Cool hard-boiled eggs in ice water. The cold water will stop them from cooking and help the eggs to contract from the membrane, thus making them easier to peel.
Yield
Makes 9 servings. A serving is 2 prepared egg halves.