Place beet and 1 1/2 teaspoons salt in a saucepan of water; bring to a boil. Reduce heat to medium; boil until beet is tender, about 30 minutes. Drain and peel beet; grate on large holes on a box grater, and place in a bowl. Grate carrot; add to beets.
Cut fennel into very thin 1/2-inch-long slivers; add to beets and carrots.
Whisk together vinegar, olive oil, pepper, and remaining 1/4 teaspoon salt. Toss with vegetables. Refrigerate for 1 hour.
Trim endive, separating leaves. Reserve large leaves for another use. Fill lower half of small leaves with 1 1/2 teaspoons of vegetable mixture. Top with 1/8 teaspoon salmon roe, if using; garnish with chervil.