Pre heat the oven to 400 degrees for the baking of the chips. Rinse and slice the tomato into ¼' inch pieces. Rinse and half the lemon. Rinse the celery and small dice half. Cut the other half of the celery into 1' sticks. Rinse and thinly slice the green onions, saving the white parts of the onions for the poaching liquid. Rough chop the walnuts. Rinse and rough chop the parsley. Rinse the potato and thinly slice into potato chip sized slices.
Bake the Chips
In a mixing bowl, combine the potato chips, 2 tsp. of olive oil, and season with a pinch of salt and pepper. Mix well to incorporate. Place the chips on a foil lined baking sheet and cook for 10 minutes on each side or until golden brown and crispy.
Poach the Chicken
Rinse the chicken and place in a medium sized pot. Cover the chicken with water. Add the juice and fruit of half the lemon, white parts of the green onions, and the celery sticks. Season the liquid with a pinch of salt and pepper. Bring to a boil and reduce to a simmer. Cook for 10-12 minutes or until firm to the touch and no pink remains. Internal temperature of 165 degrees. Place the chicken into a bowl and cool in the refrigerator before shredding.
Shred the Chicken
Take the chicken and shred in the bowl with a fork.
Mix the Chicken Salad
In the mixing bowl with the shredded chicken, add the mayonnaise, chopped celery, green onions, walnuts, parsley, juice of the remaining half of lemon. Mix well to combine, and season with a pinch of salt and pepper. Refrigerate until serving.
Place two buns on the plate. Top the buns with the chicken salad, garnish with sliced tomato and arugula. Serve with potato chips on the side of the sandwiches.