Melt the ghee in a skillet over medium heat. Add the mushrooms, onion, garlic, and arugula, and cook until the veggies just begin to soften and greens are wilted (about 3 minutes).
Transfer to a large bowl, and allow to cool. Stir in the remaining ingredients (turkey, egg whites, salt & pepper).*
Divide the mixture & shape into 4* equally sized patties. I pinch with my fingers to make them a little thinner in the middle (helps keep them from getting too round when they cook). Sprinkle each side with salt & pepper.
Heat a large skillet to medium-high heat, and coat with cooking oil of your choice (I like avocado oil). Add all four patties to the skillet and cook for 6 minutes. Flip the burgers, and continue cooking for 4 minutes.
Serve hot!