Preparation Using hands, gently mix first 8 ingredients in large bowl just until blended. Form into four 3/4-inch-thick oval patties. (Can be made 1 day ahead. Cover; chill.) Prepare barbecue (medium-high heat). Grill patties until cooked to desired doneness, about 6 minutes per side for medium-rare. Transfer patties to plates. Top with avocado slices and Lime-pickled Red Onions. Serve, passing salsas alongside.