In the work bowl of a food processor, pulse ham until finely ground.
In a large bowl, combine ham, mayonnaise, eggs, celery, mustard, pickle relish, and pepper, stirring to blend.
Using a 3-tablespoon levered scoop to portion evenly, divide ham mixture among bread slices, spreading to cover in a thick layer. Top ham layer of 5 bread slices with a layer of lettuce. Place remaining bread slices atop lettuce, ham side down.
Using a serrated bread knife, trim crusts from sandwiches, creating even squares. (Discard crusts.) Cut each sandwich diagonally into quarters, creating 4 triangles.
Recipe by TeaTime Magazine at https://teatimemagazine.com/ham-and-egg-salad-tea-sandwiches-recipe/