French Onion Deviled Eggs

1. Heat oil in a medium-size nonstick skillet over medium-low. Add onion and 1/2 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.

2. 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.

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

4. 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.

5. 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.

6. 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 1/4 tsp. salt. Stir well to combine.

7. 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.

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. Use immediately, or store in an airtight container at room temperature up to 3 days.