Boil the eggs: Place the eggs in a single layer in a large saucepan and cover with cold water by about an inch. Bring to a full boil over high heat.

Rest off the heat: As soon as the water boils, turn off the heat and cover the pot. Let the eggs sit for 11 minutes so the yolks set fully without a gray ring.

Cool in an ice bath: Transfer the eggs to a bowl of ice water and let them sit for 5 minutes. This stops the cooking and loosens the shells for easier peeling.

Peel the eggs: Tap each egg gently all over on the counter, then roll it under your palm to crack the shell evenly. Peel under cool running water to remove any stubborn bits.

Halve and separate yolks: Slice each egg in half lengthwise with a sharp knife. Scoop the yolks into a bowl and set the whites cut side up on a platter.

Mash the yolk filling: Mash the yolks with 1/4 cup mayonnaise, mustard, vinegar, salt, pepper, and 1/4 teaspoon paprika until smooth and creamy, with no lumps remaining.

Fill the egg whites: Spoon or pipe the yolk mixture into each egg white half, mounding it slightly above the rim so there's enough surface for the web.

Thin mayo for piping: Whisk the remaining 1/4 cup mayonnaise with the water until it drips slowly off a spoon. This thinner mayonnaise will form the web lines.

Pipe the web rings: Spoon the thinned mayonnaise into a zip-top bag and snip a tiny corner. Pipe a small dot in the center of each filled egg, then pipe 3 to 4 thin rings around it.

Drag into a web pattern: Drag a toothpick from the center dot out to the edge in 6 to 8 straight lines, wiping the toothpick clean between each drag. The rings will pull into pointed web spokes.

Add the spider legs: Arrange thin olive slices around the edge of each egg, bending them slightly to look like spider legs crawling onto the web.

Garnish and chill: Dust the eggs lightly with the remaining paprika and scatter chives over the platter. Chill for 15 minutes before serving so the web holds its shape.