Heat 2 tsp. oil in a large heavy ovenproof skillet over medium heat. Add onions and sugar and cook for 5 minutes. Reduce the heat to low and continue cooking, stirring often, until the onions are deeply colored and caramelized, 7 to 10 minutes. Add wine, chicken stock, currants, capers and rosemary. Simmer until reduced to 1 cup, about 15 minutes. Transfer the sauce to a small bowl and set aside. Rinse and dry the skillet. (this sauce can be prepared and refrigerated up to 2 days ahead of time).

Preheat oven to 400 degrees F. With a sharp knife, split each quail down one side of the backbone. Turn over and press to flatten. Remove any excess fat. Lightly season both sides of the birds with salt and pepper. Heat the remaining 2 tsp. oil in the skillet over high heat. Brown the Quail, breast-side down, for 4 minutes. Turn and brown other side for 4 minutes more. Transfer to a plate. Return the sauce to the skillet and bring to a simmer. Add the quail, breast-side up, and baste with sauce. Transfer the skillet to the oven and roast for about 15 minutes, or until the thigh juices run clear when pierced with a knife.
Remove the skillet from the oven and transfer the quail to warm platter. In a small bowl, dissolve the cornstarch in 2 tsp. water and whisk into the sauce. Bring sauce to a simmer, stirring constantly, until slightly thickened, about 1 minute. Season with salt and pepper to taste. To serve, pour the sauce over the quail and surround with Wilted Spinach.