Heat 1/4 cup of the olive oil in a large nonstick pan over medium-high heat. Add the potatoes and cook for 15 - 20 minutes, stirring often, until the potatoes are soft. Transfer the cooked potatoes to a plate.
Crack the eggs into a large bowl. Add the salt. Whisk until combined. Add the potato mixture to the eggs and stir until combined.
Heat the remaining 1/4 cup olive oil in the pan over medium heat. Pour the potato/egg mixture into it. Cook for 5 - 8 minutes, until the the bottom is golden brown. Place a large plate over the pan and carefully flip the tortilla. Gently slide the tortilla back into the pan. Cook for 5 - 8 more minutes, until golden brown.
Place the cooked tortilla espanola on a cutting board and cut into bite-sized pieces. Serve with garlic aioli for dipping.