Preheat the oven to 400 degrees F.
Rinse the chicken under cold water and pat dry. Slip your fingers under the breast skin and gently create a pocket the length of both breasts. Season the inside of the cavity with 1/2 teaspoon each of salt and pepper. Set aside.
Add the arugula and 5 cloves of the garlic to a food processor fitted a blade. Pulse 3 to 4 times, then add 1/2 teaspoon salt and pepper, squeeze in the juice of 1/2 lemon and pulse 3 to 4 more times. Add the cheese and pistachios, and with the processor running, drizzle in the olive oil. Scrape down sides and pulse a few more times. Adjust seasoning, if necessary.
Gently work half of the pesto under the breast skin. Reserve the other half of the pesto. Put the other lemon half into the cavity along with the remaining 2 peeled garlic cloves. Rub the chicken all over with 1 tablespoon of the butter. Season the outside of the chicken with the remaining salt and pepper. Tuck the wing tips under the bird, and truss the chicken. Put the chicken, on its side, into a roasting pan and put in hot oven. Roast at 400 degrees F for 15 minutes, rotate chicken onto other side, and roast for 15 minutes more. Carefully turn chicken onto its back (breast side up), reduce heat to 350 degrees F and roast for 30 minutes. After 30 minutes, begin basting it with the pan juices. Repeat until chicken is done and an instant-read thermometer registers 165 degrees F.
Remove the roasting pan from the oven, put the chicken on a carving board and tent it with foil. Remove the fat on the top of the pan juices. Put the roasting pan over a burner on medium heat and add 1/3 cup of the reserved pesto. Stir and incorporate it into the pan juices along with any bits that might be on the bottom of the pan. Whisk the cornstarch and the half-and-half in a small bowl and slowly add it to the pan. Stir continually until the mixture starts to thicken. Adjust seasonings, if necessary, and pour it into a serving bowl.
Carve the chicken, arrange it on a serving platter and serve with the sauce.