Place the eggs in a single layer in a saucepan and cover with cold water by about 1 inch. Bring the water to a rolling boil over high heat. As soon as the water boils, cover the pot, turn off the heat, and let the eggs sit for 12–13 minutes.
While the eggs cook, melt the butter in a skillet over medium heat. Add the finely chopped shallots and mushrooms. Cook, stirring occasionally, until the mushrooms are browned and tender, and any moisture has evaporated, about 7–8 minutes. Remove from heat and let cool.
When the eggs are done, drain them and transfer immediately to an ice bath or run under cold water until cool to the touch. Gently tap and peel the eggs.
Slice the eggs in half lengthwise. Remove the yolks and place them in a bowl. Arrange the egg white halves on a serving plate, hollow side up.
To the bowl with the yolks, add the mayonnaise, apple cider vinegar, salt, and pepper. Mash and mix with a fork until smooth. Stir in the cooled mushroom and shallot mixture until well combined.
Spoon or pipe the filling evenly into the egg white halves. Sprinkle with chopped chives to garnish and serve immediately.