Heat olive oil in a small heavy saucepan over medium heat; saute onion until golden, 5 to 7 minutes. Add bulgur to pan along with 1 cup of water or broth. Cover and simmer on low until water is absorbed, about 15-18 minutes. Put cooked bulgur in a large mixing bowl with remaining ingredients and mix well. Form into 4 patties, about 1/2 cup of mixture per patty. Chill at least 20 minutes.
While patties chill, prepare aioli by pulsing all ingredients in a blender or food processor until blended. Chill until ready to use. In a large skillet over medium heat, add 1 Tbsp. oil. When hot, drop in patties and cook until golden, about 5 minutes per side.
Serve on whole wheat rolls with Lime-Cilantro Aioli and desired toppings (lettuce, tomato, red onion, avocado).