Preheat oven to 425°F. Slice potato crosswise into 4 (½-inch-thick) rounds. Toss potato slices with oil and ⅛ tsp each salt and pepper. Place in a single layer on a rimmed baking sheet.
Bake 20 minutes, turning potatoes after 10 minutes.
Meanwhile, cook patties in a grill pan or cast-iron skillet over medium-high heat 5 to 6 minutes per side or until done.
Mix together avocado and ⅛ tsp each salt and pepper in a bowl, mashing slightly with a fork.
Place 2 potato rounds on each of 2 plates. Top with lettuce, burgers, mashed avocado, onion, tomato, and cilantro.