Season short ribs all over with salt.

Heat 2 Tbsp. oil in a small pot over medium-high heat.

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 browned ribs to a plate.

Add onion, garlic, and ginger to the 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, 3–3 1/2 hours.

Add radish about 1 hour before ribs are done.

Remove from heat.

Heat remaining 1 tsp. oil in a large nonstick skillet over medium heat.

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 egg cylinder to a cutting board and thinly slice into ribbons.

Top ribs with egg, shilgochu, and scallions.

Serve with rice alongside.