Combine egg and yogurt and add breadcrumbs. Allow to soak in the refrigerator while grinding the meat, if possible.
Prepare the sauce by stirring together all ingredients. Set aside.
Prepare a charcoal grill for indirect cooking over medium-low heat (about 350–375°). Gently combine the ground meats with the soaked breadcrumbs, shallots, garlic, thyme, salt and pepper.
Form into a loaf about 6 inches wide and 9 inches long. Place loaf on a perforated grill pan or a sheet of foil and transfer to the grill away from the coals for indirect cooking.
Top the meatloaf with half of the sauce. Close the grill and cook for 45 minutes to 1 hour, until the internal temperature reaches 155°. Allow the loaf to rest for 10 minutes and then slice to desired thickness and serve on bread with reserved sauce. Add arugula and grilled red onions if desired.