Step 1

Soak rice cakes in a medium bowl of cool water at least 1 hour and up to 12 hours. Drain and pat dry with a kitchen towel; set aside.

Step 2

Using a slotted spoon, carefully lower eggs one at a time into a small saucepan of boiling water. Cook, adjusting heat to maintain a gentle boil as needed, 6½ minutes. Transfer eggs to a bowl of ice water and let sit until just slightly warm, about 2 minutes. Peel eggs, cut in half, and set aside.

Step 3

Cook butter in a small saucepan over medium heat, occasionally swirling pan, until butter foams then browns, 3–5 minutes. Remove from heat and immediately whisk in garlic, gochujang, gochugaru, vinegar, salt, and 2 Tbsp. water. Reduce heat to low and keep sauce warm.

Step 4

Arrange about 3 cups arugula on a platter or large plate.

Step 5

Heat oil in a large skillet over medium-high. Cook rice cakes in a single layer, undisturbed, pressing down occasionally with a spatula to ensure they are making good contact with pan, until golden brown and crisp, about 5 minutes. Turn over and cook until golden brown on the other side, about 5 minutes. Transfer rice cakes to platter with arugula; reserve skillet.

Step 6

Reduce heat to medium and cook Halloumi cheese in reserved skillet until deep golden brown, about 30 seconds per side. (It’s okay if some pieces break apart.)

Step 7

Arrange cheese on platter with arugula and rice cakes. Top with remaining arugula and reserved jammy eggs. Spoon warm sauce over.