Place raw eggs in their shells in a large pot and cover them with enough cold water to cover the eggs by two inches. Place the pot on your stove over medium-high heat and bring it to a boil, uncovered.
When the water is boiling rapidly, turn off the heat, cover the pot, and let the eggs sit in the water for 15 minutes. Remove the eggs with a slotted spoon and place them in a bowl of ice water to cool.
Place your eggs on the steamer rack inside the Instant Pot. Add one cup of hot water to the pot, cover with the lid and seal it.
Pressure cook on high for 5 minutes. (Use the egg setting if your machine has one.) After five minutes of cooking, set a timer and allow the Instant Pot to do a natural release of steam for five minutes.
After five minutes of natural release, wear oven mitts and use a long wooden spoon to press down the spout to do an instant release of the remaining steam.
Remove the eggs with a slotted spoon and place them in a bowl of ice water to cool.
While your eggs are cooking, start making your filling. In a small mixing bowl, add your pesto, mayonnaise, vinegar and the zest of one lemon. Stir to combine.
When the hard-boiled eggs are cool, peel them under cold running water and discard the shells. Pat the eggs dry.
On a cutting board, slice the eggs in half the long way. To remove the yolks, gently squeeze each egg half over your mixing bowl so the yolk drops into the bowl with the rest of the filling.
Mash the egg yolks with a fork and stir them into the rest of the filling until well combined and there are no lumps.
Use a small scoop to add some filling to each egg white half. If you don't have a scoop, use a regular teaspoon as a scoop and another spoon to scrape off the filling into the egg whites. Sprinkle on paprika for garnish.
Chill pesto deviled eggs until serving them. Do not leave deviled eggs out for more than two hours. Refrigerate leftovers, covered, for up to one week.