Cover eggs with cold water by 1 inch in a saucepan. Boil, cover, remove from heat, and rest 11 minutes. Chill in ice water 5 minutes.

Peel and coarsely grate eggs into a bowl. Fold in yogurt, Dijon, 2 teaspoons lemon juice, salt, and pepper, retaining egg texture.

Broil only cut bagel faces 1 to 2 minutes; cool on a rack.

Halve and pit avocados; scoop and thickly slice flesh. Gently toss with remaining teaspoon lemon juice.

Divide avocado among bottoms; layer egg mixture evenly on top. Close, halve, and serve promptly or refrigerate until eating.