In a medium bowl, stir together chicken, celery, arugula, 4 tablespoons mayonnaise, chutney, mustard, salt, and pepper until combined.
Spread remaining 2 tablespoons mayonnaise in a thin layer onto bread slices. Spread half of chicken salad onto mayonnaise side of 3 swirl bread slices. Top each with a rye bread slice, mayonnaise side up. Spread remaining half of chicken salad onto mayonnaise side of rye bread slices. Cover with remaining swirl bread slices, mayonnaise-side down, to make 3 triple-stack sandwiches.
Using a serrated bread knife in a gentle sawing motion, trim and discard crusts from sandwiches. Cut each sandwich in half vertically and cut each half diagonally into 2 triangles. Serve immediately, or cover with a damp paper towels, place in a covered container, and refrigerate until needed, up to 1 hour.
Recipe by TeaTime Magazine at https://teatimemagazine.com/chicken-chutney-tea-sandwiches-recipe/