Slice the injeolmi into finger-width strips if it isn't pre-cut. If it's been in the fridge, it will be firm — that's fine, it softens the moment it hits heat.
Melt 1 tbsp butter in a nonstick pan over medium-low heat. Lay the rice cake strips in a single layer and fry, turning every minute or so, until each side is lightly browned and the strips have gone soft and glossy, about 4 to 5 minutes total.
While the strips are still hot, toss them in a bowl with the 2 tbsp soybean powder until every side is coated. Set aside.
Wipe out the pan. Beat the cream cheese with 1 tbsp honey and the pinch of salt until smooth.
Butter one side of each bread slice. Spread the cream cheese mixture on the unbuttered side of two slices.
Pile the coated rice cake onto the cream cheese, drizzle with a little more honey, and close each sandwich with the second slice, buttered side out.
Toast in the same pan over medium-low heat, pressing gently with a spatula, until both sides are deep golden and the rice cake inside has gone soft and stretchy again, 2 to 3 minutes per side.
Slice in half while it's hot — this is when the rice cake pulls in long strings, which is the entire point of eating it warm.