Warm the 2 tablespoons of olive oil in a large (12-inch diameter) nonstick pan over medium-high heat. Add the onions and cook until translucent, about 4 minutes. Add the garlic and cook until tender and fragrant, about another 2 minutes. Season the onion mixture with salt and pepper. Transfer the onion mixture to a large bowl and set aside. Reserve the pan.
Meanwhile, grate the potatoes in a food processor using the grating attachment blade. Use a kitchen towel to help squeeze out the water from the grated potatoes. Add the potatoes to the bowl with the onion mixture. Add the Parmesan and the basil. Stir to combine and season with additional salt and pepper, to taste.
Warm the remaining olive oil over high heat in the same pan that was used to cook the onions. When the pan is hot, but not smoking add the potato mixture. Use a spatula to press the mixture down into the pan firmly and evenly. Turn the heat down to medium and cook the potato mixture until the bottom is golden brown and the pancake can move in the pan, about 12 to 15 minutes. Turn the heat down to medium-low if the pancake is browning too fast in places. Place a large plate on top of the pancake and flip the pancake out of the pan. Turn the heat on the pan back up to high. When the pan is hot slide the pancake back into the pan and cook until the bottom is golden and is cooked through, about 12 to 15 minutes. Slide the pancake onto a serving platter, slice, and serve.