Slice potatoes into 1/8" rounds. Core tomato and cut into 1/3" rounds. Toss potatoes on baking sheet with 1/2 tsp. salt and 2 Tbsp. olive oil, rubbing oil into potatoes with your hands. Spread into a single layer and roast 20 minutes. Remove from oven, flip, and roast 15-20 more minutes, or until potatoes brown on edges. Some potatoes may brown before others, so watch closely and remove individual potatoes early if needed.

Rinse chicken breasts and pat dry. On a separate cutting board, halve chicken breasts by cutting vertically. Lay four chicken halves on cutting board and cover with plastic wrap. Using a meat mallet or small pan, pound chicken to an even 1/3" thickness. Season chicken on both sides with 1/4 tsp. salt and 1/4 tsp. pepper.

Heat canola oil in a large non-stick pan over medium heat. (Alternatively, use a medium non-stick pan and work in batches.) Place half the ranch dressing in a mixing bowl (reserve remaining for topping sandwich). Place breadcrumbs in a second mixing bowl. Dip chicken in ranch dressing and turn to coat. Transfer to breadcrumbs and toss to coat. Press crumbs to adhere them to chicken. Transfer breaded chicken to a plate.

Line a plate with a paper towel. Test oil temperature by adding a pinch of breadcrumbs to it. It should sizzle gently. If it browns immediately, turn heat down and let oil cool. If it doesn't brown, increase heat.

Add chicken pieces to pan in a single layer (work in batches as needed), and cook 2-4 minutes per side, or until golden brown and chicken reaches a minimum internal temperature of 165 degrees. Remove to towel-lined plate. Place buns directly on oven rack and bake 3-4 minutes, or until toasted.

Assemble sandwiches in the following order: bottom bun, half the remaining ranch dressing, first chicken cutlet, cheese slice, second chicken cutlet, arugula, tomato, pickle chips, remaining ranch dressing, and top bun. Serve chips on side.