Place eggs in a single layer in a large Dutch oven. Add enough cold water to cover the eggs by at least 1 inch. Bring to a rapid boil over high heat (water will have large rapidly breaking bubbles). Remove from heat. Cover and let stand for 15 minutes; drain. Run cold water over the eggs or place them in ice water until cool enough to handle; drain. Peel eggs and cut in half lengthwise. Remove yolks and place in a large bowl. Set whites aside.
Mash egg yolks with a fork. Add mayonnaise, mustard, caper juice, black pepper, and hot pepper sauce, stirring to mix.
Chop avocados into 1/2-inch pieces; toss with lemon juice. Spoon or pipe egg yolk mixture into egg white halves. Place pieces of avocado on top. If desired, sprinkle with chives. Cover and chill for up to 4 hours before serving.