P. Terry's Burger Stand. Multiple locations but my favorite fast food cheeseburger in the USA. Walton's Fancy and Staple. Delicious lunch and pastry options. June’s All Day (South Congress).. Best cheeseburger in Austin and maybe the United States. Word of Mouth Deli/Bakery (2 locations). Get the herbed beef sandwich. Swedish Hill (West 6th).. Good for a sit down breakfast.