Preheat oven to 450°. Line a sheet-pan with foil and place a baking rack over half of the sheet pan. Set it aside. In a bowl, season chicken with parsley, 1 Tbsp garlic, oil, lemon zest and salt to taste.
In a medium size bowl, toss shallots, broccolini, grape tomatoes and fingerling potatoes with oil. Set it aside.
Place the potatoes scattered on the side of the prepared sheet pan without the roasting rack. Roast for about 5 minutes. Remove from oven, quickly place the chicken over the rack, return to oven and roast for 10 minutes.
In the meantime, in a small bowl, combine butter, lemon juice, wine, capers, parsley, the remaining 1Tbsp garlic, and season with a pinch of salt and freshly ground pepper. Set it aside.
Remove sheet pan from oven and scatter the broccolini rabe, tomatoes, and shallots in the pan, if needed move the potatoes a bit under the chicken, rotate the pan and roast for 5 minutes.
Remove from the oven, remove the baking rack, placing the chicken over everything; then pour the butter mixture over everything and roast for 5 more minutes.
Remove from oven and plate your Piccata Chicken with its sides; garnish with Parmigiano-Reggiano, extra parsley, and drizzle the pan juices over, thin slices of lemon.