Boil the potato with its peel on until tender. Drain, peel, and mash through a food mill or potato ricer.
Mash the tuna in a bowl. Add the grated cheese, the whole egg plus the white, pepper, and the mashed potato. Mix into a homogeneous mixture.
Moisten cheesecloth, wring until just damp, and lay it flat. Place the tuna mixture at one end and shape into a roll about 2 1/2 inches in diameter. Wrap three or four turns and tie each end securely.
Put the onion, celery, carrot, parsley stems, a pinch of salt, and the wine in a pot. Add the roll and enough water to cover it by at least 1 inch. Cover and bring to a boil, then reduce to the gentlest simmer and cook 45 minutes.
Lift the roll out carefully from both ends, unwrap as soon as you can handle it, and let cool completely.
Make the mayonnaise as directed, then mix in the capers and anchovy.
Slice the cold roll into pieces less than 1/2 inch thick and arrange on a platter, overlapping slightly. Spread with mayonnaise and garnish with slivered olives.