In a large bowl, whisk together 2 large eggs and 2 egg yolks until well combined. Stir in 100g of finely grated Parmigiano Reggiano and a generous amount of freshly ground black pepper. Set aside.
Bring a large pot of water to a boil and add a pinch of salt. Add 400g of Korean rice cakes (tteok) and cook until they float to the surface and are tender, about 2-3 minutes. Reserve 1 cup (240ml) of the cooking water before draining the tteok.
While the tteok is cooking, place 180g of guanciale, cut into 0.5cm (⅕") thick batons, in a non-stick pan over medium-high heat. Cook for 4-5 minutes until golden and crispy. The guanciale will render its own fat, so no additional oil is needed. In the last minute of cooking, add 2 minced garlic cloves and sauté until fragrant.
Add the drained tteok to the pan with the guanciale and garlic. Toss to coat the tteok in the rendered fat.
Remove the pan from heat to prevent the eggs from scrambling. Transfer the tteok and any residual fat into the bowl with the egg mixture. Add ½ cup (120ml) of the reserved tteok cooking water. Stir vigorously for about 1 minute until the sauce becomes creamy and clings to the tteok. If the sauce is too thick, add more reserved cooking water as needed.
Transfer the Carbonara Tteokbokki to warm bowls. Garnish with additional grated Parmigiano Reggiano, a pinch of freshly ground black pepper, and fresh thyme leaves, if desired. Serve immediately.