Peel and coarsely grate 4 medium potatoes into a large bowl of cold water to prevent browning. Let sit for 5 minutes, then drain thoroughly and squeeze out excess moisture using a clean kitchen towel.

In a large mixing bowl, combine the drained potatoes, 1 julienned carrot, 2 thinly sliced scallions, and 2 cloves of minced garlic. Add 1 beaten egg, 2 tablespoons of flour, 1 teaspoon of salt, and 1/2 teaspoon of black pepper. Mix well until all ingredients are evenly distributed.

Prepare the dipping sauce by mixing 3 tablespoons of soy sauce, 1 tablespoon of rice vinegar, 1 teaspoon of sesame oil, 1/2 teaspoon of minced ginger, and a pinch of red pepper flakes in a small bowl. Set aside.

Heat 3 tablespoons of vegetable oil in a large non-stick skillet over medium-high heat. Once the oil is shimmering, add about 1/2 cup of the potato mixture to the pan and flatten it into a round pancake about 1/2 inch thick.

Cook for 4-5 minutes on the first side until golden brown and crispy. Carefully flip the pancake and cook for another 4-5 minutes on the other side. The edges should be deeply golden and crispy while the center is cooked through.

Transfer the cooked pancake to a paper towel-lined plate to absorb excess oil. Repeat with the remaining mixture, adding more oil to the pan as needed. You should get about 6-8 pancakes depending on size.

Serve the pancakes hot, cut into wedges if desired, with the dipping sauce on the side. Garnish with additional sliced scallions and sesame seeds if desired.