Poach 1 chicken breast in a small saut or saucepan in enough water to cover by an inch. (Bring to boil, then reduce and barely simmer, partially covered, 10 minutes. Remove from heat and let sit in water 10 minutes.) When cool enough to handle, cut into chunks and pulse in processor until finely chopped. This should yield about 1/2 cup.
In a medium-size bowl, combine chicken with 1/3 cup or so diced apple, 1/4 cup mayo, 1 teaspoon lemon juice (optional), and salt and pepper to taste. (This is basic version; can add any desired seasonings.) Stir until thoroughly combined.
To make 1 roll: trim crusts from 1 slice sandwich bread (need spongy supermarket-style bread here; not artisanal-style with big or coarse crumb). Using a rolling pan, roll slice out a bit, just to flatten and soften. Lightly spread with about 1/2 teaspoon mayo, then thin layer of chicken salad (about 2 tablespoons). Top with single layer of julienned carrots (thin shavings using julienne peeler) and cucumber. Starting at one short end, roll up like a sushi roll. Repeat with 3 more slices of bread to make 4 rolls total. Chill for 10 minutes; then cut crosswise into smaller rolls.