Using a square of aluminum foil, create a cup to hold the garlic, root side down. Drizzle a generous portion of oil over the garlic and lightly sprinkle with salt. Twist the aluminum cup closed and roast the packet in the oven for 8-10 min, or until the garlic is slightly squishy. Set aside for later use.
Meanwhile, wash & prep the sprouts, removing the woody ends, then cut in half. If sprouts are large, you may want to cut in quarters.Remove wrinkled or yellow outer leaves; pat dry (you will be putting these in hot oil).
Thinly slice the onion into rings; set aside.
In a large sauté pan, heat the ¼ cup of oil over medium high heat, then gently add the sprouts and onions to the hot oil. Add half of the butter (or duck fat) to keep the sprouts nicely green & shiny. As the sprouts start to crisp (about 5-6 min.), add remaining butter or duck fat.
Open the roasted garlic and, holding it by the foil cup, gently cut off the top of the garlic, then squeeze the contents into the pan. Season with salt & pepper to taste.
Reduce heat to medium and continue to stir until all ingredients are incorporated.
The sprouts are ready when you can easily squeeze one with a pair of tongs (15-18 min. start to finish); they should be slightly crunchy on the outside and soft inside.
Serve with a garnish of shaved hard Italian cheese.