Step 1

Season short ribs all over with salt. Heat 2 Tbsp. oil in a small pot over medium-high. Working in 2 batches, cook ribs, turning occasionally and reducing heat if needed to prevent scorching, until browned all over, 10–12 minutes per batch. Transfer to a plate.

Step 2

Add onion, garlic, and ginger to same pot and cook, stirring often, until softened and lightly browned, 6–8 minutes. Add wine; bring to a boil. Reduce heat and simmer until liquid is reduced by half, 8–10 minutes. Add mirin, soy sauce, brown sugar, and 2 cups water. Return ribs to pot and bring liquid to a simmer. Partially cover pot and cook, reducing heat to maintain a very gentle simmer and adding splashes of water as needed, until ribs are very tender (they should shred easily) and sauce is thick enough to coat meat, 3–3½ hours. Add radish about 1 hour before ribs are done. Remove from heat.

Step 3

Heat remaining 1 tsp. oil in a large nonstick skillet over medium. Add egg, tilting pan to create a very thin 6–8" circle. Cook just until set, about 1 minute, then roll up egg into a cylinder; transfer to a cutting board. Thinly slice into ribbons.

Step 4

Top ribs with egg, shilgochu, and scallions. Serve with rice alongside.

Step 5

Do Ahead: Short ribs can be made 3 days ahead. Let cool; cover and chill.