MARINATE THE GOOSE: Stir together the marinade ingredients in a large mixing cup or bowl. Place goose breasts in a non-reactive container (glass or stainless steel) and pour the marinate over them. Cover and refrigerate 6 hours or overnight.
MAKE THE VINAIGRETTE AND PREHEAT THE OVEN: About a half hour before serving, whisk together the vinaigrette ingredients in a large mixing bowl and set aside. Place rack at center of oven and preheat to 375 degrees F.
COOK THE GOOSE: Heat the butter in a large, heavy ovenproof skillet over medium-high flame until foaming. Remove goose from marinade and pat dry. Remove as much silver skin as possible with a small, sharp knife. Salt and pepper the goose breasts, place them in the pan, and sauté 3-4 minutes on each side. Place pan in oven and roast for several more minutes until an instant read thermometer reads 135 degrees F. Remove pan from oven. Transfer the goose to a plate on the stove and cover loosely with foil while you make the sauce and toss the salad. The temperature of the breasts will rise as they rest. When the sauce and salad are both ready, slice the goose breasts thinly at an angle.
MAKE THE SAUCE: Pour the water and wine into the skillet in which you cooked the goose and add the whole garlic cloves. Heat, stirring, over medium-high flame and cook a few minutes until reduced by about half.
ASSEMBLE THE SALAD: Add the arugula to the bowl with the reserved vinaigrette. Season with salt, peppers, and Pecorino Romano cheese. Toss well. Divide the salad between four shallow bowls or plates and layer ¼ of the sliced goose down the center of each. Drizzle the sauce over the meat and garnish with the toasted pine nuts. Serve immediately.