Heat oil in a medium-size nonstick skillet over medium-low. Add onion and ½ tsp. of the salt; cook, stirring occasionally, until deeply browned and caramelized, 30 to 35 minutes, adding a splash of water if onion starts to burn. Remove from heat, and stir in vinegar. Transfer onion to a small bowl; let cool completely, about 20 minutes.

While onion cools, fill a large pot with water, and bring to a boil over high. Carefully lower eggs into boiling water; cook, undisturbed, for 11 minutes, 30 seconds.

Meanwhile, fill a large bowl halfway with ice, and add enough water to cover ice; set ice bath aside.

Using a spider or slotted spoon, immediately transfer cooked eggs from boiling water to ice bath. Let stand until completely cooled, at least 5 minutes or up to 30 minutes.

Working with 1 egg at a time, firmly tap eggs on countertop or a flat work surface until cracks form all over the shell. Peel eggs under cold water.

Cut eggs in half lengthwise, and carefully remove yolks. Using a fork, mash together yolks, mayonnaise, and mustard in a medium bowl. Add reserved onion and remaining ¼ tsp. salt. Stir well to combine.

Spoon egg yolk mixture evenly into egg whites. Top evenly with Crispy Gruyère Onions. Serve immediately, or store in an airtight container in refrigerator up to 3 days.

For Crispy Gruyère Onions: Toss together 2 Tbsp. each shredded Gruyère cheese and crispy fried onions in a small bowl. Spread on a parchment-lined baking sheet. Bake at 350°F until melted and golden brown, 8 to 10 minutes. Let cool 10 minutes. Discard any burned bits of onion. Break into small pieces.