The frittata can be made in advance and served at room temperature.
Preheat oven to 425 degrees. Bring potato, 2 1/2 cups water, and 1/2 teaspoon salt to a boil in a medium nonstick skillet. Reduce to a simmer and cook until fork-tender, about 7 minutes. Drain.
Wipe out skillet; return to medium-high heat. Add oil, onion, and sausage; cook, stirring occasionally, until sausage is browned, about 4 minutes. Reduce heat to medium and add potatoes. Whisk together eggs and milk; season with salt and pepper. Pour into skillet; cook, stirring with a spatula to create large curds, 30 seconds. Transfer to oven and bake until eggs are just set, 8 to 10 minutes. Let stand 10 minutes, then slice into 8 wedges. Divide mayonnaise among bread slices, then sandwich frittata wedges, lettuce, and tomato with them; serve.