Prep the potato: Peel the potato and shred it using a food processor. Squeeze out the excess liquid using cheesecloth.
Cook the garlic, potato, egg: Heat olive oil in a large pan over medium heat. Add garlic and cook until it browns, about 1 minute. Add shredded potatoes and salt. Crack the egg over the pan and scramble it, mixing with the shredded potatoes. Flatten with a spatula and cook until the potatoes are browned, about a few minutes per side. Turn off the heat.
Assemble the quesadilla: Place a tortilla on a large pan over low heat. Add layers of arugula leaves, diced tomatoes, and then the potato-egg mixture. Sprinkle cheese. Place a tortilla on top. Turn the heat up to medium and press down with a spatula. Heat the bottom tortilla until it is crispy and brown. Flip the quesadilla and repeat for the other side. Cut into slices and serve immediately.