Preheat the oven to 375°F. Place a large cast iron skillet over high heat for 5 minutes, or until the pan is good and hot. Pat the chicken halves dry with a paper towel, then rub all over with
Roast the chicken halves for 40 minutes, or until the juices run clear when the thigh is pierced. Transfer the chicken to a platter to rest skin side up for 15 minutes reducing the oven temperature to 250°F.
When the chicken goes into the oven, peel the potatoes and cut lengthwise into ⅓-inch slices. Stack the slices and cut lengthwise into ⅓-inch fries. Place the fries in a large bowl with cold water to cover and soak for 20 minutes. Drain the water, refill, and soak for another 20 minutes. Drain the potatoes and pat as dry as possible.
Heat the vegetable oil in deep fryer or large, heavy-bottomed pot to 375°F. Working in small batches, fry the potatoes for 8 to 10 minutes, or until golden brown and tender. Transfer to a paper-towel-lined baking sheet and keep warm in the oven while you fry the remaining potatoes.
Place the parsley, thyme, tarragon, olives, capers, and anchovies on a cutting board. Chop them with a knife until they are very fine. Transfer to a bowl and stir in the lemon zest, olive oil, and vinegar. Season to taste with salt and pepper.
To serve, toss the arugula with a squirt of lemon juice and a quick glug of olive oil. Arrange the chicken halves on 2 plates with a pile of the arugula. Spoon some of the salsa verde over the chicken and serve with the fries.
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