Drain the tofu and gently press between paper towels or in a tofu press for about 15 minutes to remove excess moisture.
Cut the pressed tofu into 1/2-inch thick rectangular slices.
In a small bowl, combine soy sauce, sesame oil, gochugaru, sugar or honey, minced garlic, half of the green onions, and sesame seeds (and ginger if using). Mix well and set aside.
Heat vegetable oil in a large non-stick skillet over medium-high heat.
Carefully place the tofu slices in the hot pan in a single layer, making sure not to overcrowd the pan.
Cook the tofu for 3-4 minutes on each side until golden brown and crispy on the outside.
Lower the heat to medium-low and pour the sauce over the tofu slices.
Gently spoon the sauce over the tofu, slightly lifting the edges to let the sauce flow underneath.
Simmer for 3-4 minutes until the sauce has thickened slightly and the tofu has absorbed some of the flavors.
Remove from heat and garnish with the remaining green onions and additional sesame seeds if desired.