Using rasp, finely grate 4 of the garlic cloves into small bowl; stir in 3 tsps of the salt to make a paste.
Divide and massage garlic mixture over chicken halves; place skin-side up on rack set over baking sheet. Evenly zest lemon over chicken halves, reserving lemon for juice. Refrigerate, uncovered, for 8 to 12 hours.
Preheat oven to 500°F; line baking sheet with parchment paper. Thinly slice remaining garlic cloves; add to large bowl along with potatoes, olive oil and 2 tbsp lemon juice from reserved lemon. Sprinkle with pepper and the remaining 1½ tsp salt; toss to coat.
Arrange potatoes into a rough rectangle on prepared pan; place chicken halves on top. Roast for 35 minutes.
Sprinkle Parmesan over chicken (it’s okay if some lands on potatoes); return to oven until a thermometer inserted into thickest part of breast registers 165°F, about 10 to 15 minutes. Let rest for 15 minutes; remove chicken to cutting board before carving.
Serve with potatoes and a green salad.