Place the dried shiitake mushrooms in a bowl and cover with warm water. Set aside until softened, about 20 minutes.
Beat the egg yolks in a small bowl with a pinch of salt. Place a small skillet over medium heat. Lightly wipe with vegetable oil and reduce heat to the lowest setting.
Add the beaten egg yolks, tilting the pan to form a thin layer. Remove the pan from heat. Once the top is nearly set, flip to let the residual heat finish cooking the other side. Remove from the pan and slice into thin strips. Set aside as garnish.
Once the mushrooms are softened, squeeze to remove excess water. Discard the stems and thinly slice.
In a small bowl, combine the 1 1/2 tablespoons soy sauce, honey, sesame oil, and black pepper. Set Aside.
Drizzle the vegetable oil in a wok or large skillet over medium high heat. Once heated, add the onions. Cook, stirring occasionally, until starting to soften.
Add the garlic and cook until just fragrant, 30 seconds to 1 minute.
Add the marinated beef and cook, stirring occasionally, until starting to brown on all sides.
Stir in the mushrooms, green onions, carrots, and bell peppers.
Add the prepared rice cakes and the soy sauce mixture. Toss well to coat thoroughly.
Cook, stirring often, until the rice cakes and vegetables have softened, about 2 minutes.
Adjust seasonings to taste and serve hot topped with the egg strips and sesame seeds.